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  • Articles  (379)
  • Numerical Methods  (379)
  • Mathematics  (379)
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  • 1
    Electronic Resource
    Electronic Resource
    Stability of some boundary value methods for the solution of initial value problems (1993)
    Springer
    BIT 33 (1993), S. 434-451 
    Details
    ISSN: 1572-9125
    Keywords: 34A50 ; 65L05 ; 65L20 ; Ordinary Differential Equations ; Initial Value Problems ; Numerical Methods ; Stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01990527
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  • 2
    Electronic Resource
    Electronic Resource
    Semigroup linearization for nonlinear parabolic equations (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991), S. 147-163 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present an algorithm for nonlinear parabolic equations that uses a linear semigroup approach to decouple the nonlinearity, thus allowing simpler techniques to be used for the solution. The basic method is applicable only to a restricted class of problems, but can be extended with little loss of utility. Stability and error estimates are given, along with example computations.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690070204
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  • 3
    Electronic Resource
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    Fourth-order finite difference scheme for a system of quasilinear elliptic equations (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991), S. 179-191 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The system of two quasilinear elliptic equations is approximated by the method of lines, which has the truncation error O(h2) at points neighboring the boundary and O(h4) at the most interior points. It is proved that the global error of the method is O(h4) at all mesh points. The two-point boundary value problem for the system of ordinary differential equations that arises from the method of lines is solved by the O(h4) convergent finite difference scheme, suitable to the equations of the form uxx = f(x, u) without the first derivative ux. The system of algebraic equations obtained by the full discretization is solved by Gauss elimination method for three diagonal matrices combined with the method of iterations. A numerical example is presented.
    Additional Material: 1 Tab.
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    URL: http://dx.doi.org/10.1002/num.1690070206
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  • 4
    Electronic Resource
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    Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Part II: L∞-error estimates (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991), S. 85-99 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690070107
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  • 5
    Electronic Resource
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    Solving eigenvalue problems by implicit decomposition (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991), S. 113-145 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper presents three innovative methods for solving eigenvalue problems for differential equations based upon the techniques of implicit decomposition developed by Luo and Friedman. An eigenvalue problem can be written as an approximate algebraic system of the form [K]{X} + λ[M]{X} = 0 by employing finite elements. These methods provide robust techniques to compute the real eigenpair, λ and {X}, where [K] and [M] can be asymmetric, indefinite, and even singular.
    Additional Material: 1 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690070203
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  • 6
    Electronic Resource
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    Asymptotic factorization of evolution equations which involve linear operators (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991), S. 165-177 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Operator splitting for evolution equations which involve linear operators is investigated. A systematic method for identifying multifactor fractional-step operator splittings is proposed. The classical second-order accurate splittings of Strang readily emerge, as well as various third-order splittings of nonstandard type. Factorizations possessing mixed positive-negative fractional steps are discovered, as well as bizarre splittings whose factors include complex fractional steps. It is found that no higher-order accurate splitting which employs all positive fractional steps and which possesses less than nine operator factors can exist. For the linear case this implies second-order accurate splittings are optimal, as a further increase of one power in accuracy essentially requires a three-fold increase in work. This estimate slightly improves, for cases where the nonstandard splittings can be employed. Some numerical experiments which illustrate the theory are provided.
    Additional Material: 3 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690070205
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  • 7
    Electronic Resource
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    Numerical simulation of the turbulent boundary layer equations via a Runge-Kutta algorithm (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991), S. 375-384 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper addresses a hybrid computational procedure for the step-by-step calculation of momentum transfer in turbulent boundary layer flows along flat plates. The proposed procedure relies on a modified method of lines wherein transversal discretizations are carried out by a “control volume” being infinitesimal in the streamwise direction and finite in the transversal direction of the fluid flow. Using mixing length theory and coarse intervals in the transversal direction, the resulting system of ordinary differential equations of first order may be readily integrated on a personal computer utilizing a fourth-order Runge-Kutta algorithm. In general, a maximum number of sixteen lines is necessary at the trailing edge of the flat plate for a typical calculation. As a consequence, computing time and storage for each run were very small when compared to other finite-difference methods. Furthermore, to validate the hybrid procedure involving the method of lines and control volumes (MOLCV), comparisons with experimental data have been done in terms of both velocity distributions and local skin friction coefficients. For all cases tested, the proposed methodology predicts the growth of the boundary layer of gases correctly.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690070407
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  • 8
    Electronic Resource
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    On the dynamics of a discrete reaction-diffusion system (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991), S. 385-405 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We investigate various aspects of the dynamics of a discrete reaction-diffusion system. First, we examine the effect of the boundary conditions on the spatially uniform fixed point at locations far from the boundaries by using an asymptotic expansion. We show that, except for a few computational cells adjacent to the boundary, the fixed point practically coincides with the one calculated by using reflective boundary conditions (equivalent to an infinite domain). Next, we introduce a classification of the fixed points based on the wavelength in the infinite-medium approximation of the system. We use the symbolic manipulator MACSYMA to analytically calculate the amplitude of several such classes of fixed points and we generate bifurcation diagrams for their members. Also, we consider two special classes of periodic solutions; we calculate their amplitude analytically in the infinite-medium approximation, and generate bifurcation diagrams that shed new light on some previous confusing results. Finally, we present an analysis of fictitious periodic solutions that have been previously reported and incorrectly interpreted.
    Additional Material: 8 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690070408
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  • 9
    Electronic Resource
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    Masthead (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992) 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690080201
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  • 10
    Electronic Resource
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    Discontinuous solutions to hyperbolic systems under operator splitting (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991), S. 277-297 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Two-dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time-steps exactly solves the component one-dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to “pseudo acoustic waves” and a nonphysical stationary wave. The implications of these findings for the design of high-resolution computing schemes are discussed.
    Additional Material: 6 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690070306
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  • 11
    Electronic Resource
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    Masthead (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991) 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690070401
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  • 12
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    A message to our readers (1991)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 7 (1991), S. iii 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/num.1690070402
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  • 13
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    Numreical solutoin of a class of parabolic partial differential equation arising in optimal control problems with uncertainty (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 77-95 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper the optimal control of uncertain parabolic systems of partial differential equations is investigated. In order to search for controllers that are insensitive to uncertainties in these systems, an iterative optimization procedure is proposed. This procedure involves the solution of a set of operator valued parabolic partial differential equations. The existence and uniqueness of solutions to these operator equations is proved, and a stable numerical algorithm to approximate the uncertain optimal control problem is proposed. The viability of the proposed algorithm is demonstrated by applying it to the control of parabolic systems having two different types of uncertainty.
    Additional Material: 12 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690080105
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  • 14
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    Optimal convergence rates for the combined methods of different finite element methods (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 203-220 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Coupling techniques are essential to combining different numerical methods together for the purpose of solving an elliptic boundary value problem. By means of nonconforming constraints, the combinations of various Lagrange finite element methods often cause reduced rates of convergence. In this article, we present a method using penalty plus hybrid technique to match different finite element methods such that the optimal convergence rates in the ‖ · ‖h and zero norms of errors of the solution can always be achieved. Also, such a coupling technique will lead to an optimal asymptotic condition number for the associated coefficient matrix. Moreover, this study can easily be extended for combining the finite difference method with the finite element method to also yield the optimal rate of convergence.
    Additional Material: 3 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690080301
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  • 15
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    Mixed-hybrid finite elements and streamline computation for the potential flow problem (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 221-266 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: An important class of problems in mathematical physics involves equations of the form -∇ · (A∇φ) = f. In a variety of problems it is desirable to obtain an accurate approximation of the flow quantity u = -A∇φ. Such an accurate approximation can be determined by the mixed finite element method. In this article the lowest-order mixed method is discussed in detail. The mixed finite element method results in a large system of linear equations with an indefinite coefficient matrix. This drawback can be circumvented by the hybridization technique, which leads to a symmetric positive-definite system. This system can be solved efficiently by the preconditioned conjugate gradient method. After approximating u by the lowest-order mixed finite element method, streamlines and residence times can be determined easily and accurately by computations at the element level.
    Additional Material: 11 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690080302
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  • 16
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    Probabilistic viscosity algorithm for the scalar conservation law (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 277-290 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We derive an algorithm for solving the initial value problem for ut = ½σ2uxx + f(u)ux. The approach is based on the representation of the solution to the above equation in the form of the functional of Brownian motion. For small σ we get the approximation for ut = f(u)ux. A comparison with the random choice method is illustrated by the numerical example.
    Additional Material: 3 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690080304
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  • 17
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    Masthead (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992) 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690080401
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  • 18
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    The primal mixed finite element method and the LBB condition (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 357-379 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Spurious or kinematic modes have posed a major obstacle to the implementation of the mixed finite element method. This research shows that spurious modes resulting from the approximation spaces not satisfying the LBB condition do not prevent a well posed problem. When the LBB condition is not satisfied, the resulting matrix equations are singular. A direct solution method is presented for the efficient solution of the possibly singular equations. Orthogonal flux basis functions are introduced to simplify the problem. Then the solution procedure is based on nested domain decomposition. This solution procedure is shown to be competitive with direct solution methods for the displacement finite element method. Examples are included to demonstrate various aspects of the LBB condition and the solution procedure.
    Additional Material: 7 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690080405
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  • 19
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    Domain-decomposition approach to local grid refinement in finite element collocation (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 341-355 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Advection-dominated flows occur widely in the transport of groundwater contaminants, the movements of fluids in enhanced oil recovery projects, and many other contexts. In numerical models of such flows, adaptive local grid refinement is a conceptually attractive approach for resolving the sharp fronts or layers that tend to characterize the solutions. However, this approach can be difficult to implement in practice. A domain decomposition method developed by Bramble, Ewing, Pasciak, and Schatz, known as the BEPS method, overcomes many of the difficulties. We demonstrate the applicability of BEPS ideas to finite element collocation on trial spaces of piecewise Hermite cubics. The resulting scheme allows one to refine selected parts of a spatial grid without destroying algebraic efficiencies associated with the original coarse grid. We apply the method to steady-state problems with boundary and interior layers and a time-dependent advection-diffusion problem.
    Additional Material: 10 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690080404
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  • 20
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    Stability analysis of discrete generalized Stokes problems (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 303-323 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In the mixed finite element approximation of Stokes-like problems, stability considerations normally restrict the choice of the finite element subspaces to families satisfying the so-called inf-sup condition. However, in the case of the generalized Stokes problems that one encounters when solving either the nonstationary compressible or incompressible Navier-Stokes equations by means of some operator splitting, the contribution of the time derivative is adding a term in the continuity equations which, at least theoretically, leads to a problem that is always stable. On the other hand, the numerical solution of the discrete problems built with approximations not satisfying the inf-sup condition are generally plagued with oscillations that look pretty much like the checkerboard phenomenon occurring in the classical context. To clarify this apparent contradiction, we undertook a theoretical study of the condition number of the linear problem involving the sensitive variable, which for such a saddle point problem is the dual problem. In the course of this study, we have exhibited the relationship between that condition number and the inf-sup constant, thus providing a different point of view on the results of Brezzi and Babuska. Finally, we conducted a parametric study of the behavior of that condition number with respect to the mesh size, the viscosity, and the time step. That study showed that, even in this context, the use of “stable element” was adding to the quality of the approximation by reducing the condition number to an acceptable level.
    Additional Material: 8 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690080402
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  • 21
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    Analysis of numerical integration in p-version finite element eigenvalue approximation (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 381-394 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study the effect of numerical integration when the p-version of the finite element method is used to approximate the eigenpairs of elliptic partial differential operators. We obtain optimal orders of convergence for approximate eigenvalues and eigenvectors under a certain set of requirements on the quadrature rules employed. This is the same set of conditions that has been shown (in an earlier work) to be sufficient for the optimal approximation of the solutions of the corresponding source problems.
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    URL: http://dx.doi.org/10.1002/num.1690080406
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  • 22
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    Error estimates for a discretized Galerkin method for a boundary integral equation in two dimensions (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 405-421 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present a priori and a posteriori estimates for the error between the Galerkin and a discretized Galerkin method for the boundary integral equation for the single layer potential on the square plate. Using piecewise constant finite elements on a rectangular mesh we study the error coming from numerical integration. The crucial point of our analysis is the estimation of some error constants, and we demonstrate that this is necessary if our methods are to be used. After the determination of these constants we are in the position to prove invertibility and quasioptimal convergence results for our numerical scheme, if the chosen numerical integration formulas are sufficiently precise. © 1992 John Wiley & Sons, Inc.
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    URL: http://dx.doi.org/10.1002/num.1690080502
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    Masthead (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992) 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/num.1690080601
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  • 24
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    Modeling of multimaterial interfaces in the finite difference method (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 493-503 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This work develops a procedure for representing multimaterial interfaces in finite difference models. The boundary separating the two materials can be on or between a row of nodes. The development is validated by embedding the boundary between two regions in a single, larger region and comparing the results. The development is facilitated by the use of a physically based notation that represents the displacement approximations in terms of rigid-body rotations and strain gradient quantities that produce the displacements. Four example problems are presented.
    Additional Material: 8 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690080507
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    Numerical solution of the heat equation by infinite-order ordinary differential equations (1992)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 8 (1992), S. 505-514 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A numerical algorithm is given for solving a class of infinite-order differential equations previously discussed by the authors. These equations yield solutions to certain time-dependent boundary-value problems for the heat equation. An extrapolation process is given for increasing the speed of convergence of the sequence generated by the method. © 1992 John Wiley & Sons, Inc.
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    URL: http://dx.doi.org/10.1002/num.1690080602
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  • 26
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    Masthead (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993) 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/num.1690090301
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  • 27
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    Analysis of alternating direction collocation methods for parabolic and hyperbolic problems in two space variables (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993), S. 191-211 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: While alternating direction implicit (ADI) collocation methods have been used for several years to solve parabolic problems in several space variables, no convergence analysis has been derived for any of these methods. We formulate and rigorously analyze ADI collocation schemes applied to the inhomogeneous heat and wave equations on the unit square subject to homogeneous Dirichlet boundary conditions and appropriate initial conditions. We prove that each method is second-order accurate in time and of optimal accuracy in space in the L2 and H01 norms. Numerical experiments confirm the predicted rates of convergence. © 1993 John Wiley & Sons, Inc.
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    URL: http://dx.doi.org/10.1002/num.1690090207
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  • 28
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    Premature in-bore separation of sabot-jacketed projectiles (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993), S. 225-234 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The premature separation of saboted projectiles undergoing high-mach-number in-bore acceleration is investigated. Results from a simplified numerical model for the problem are presented. Measured short-term velocity history for a typical projectile is used to describe the motion of a nonuniformly accelerating piston. Two numerical algorithms for calculating the shocked flow in tront of the accelerating piston are presented. Mach number and pressure time histories for the flow afford corrective input to the sabot design process. © 1993 John Wiley & Sons, Inc.
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    Galerkin method for a nonlinear parabolic-elliptic system with nonlinear mixed boundary conditions (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993), S. 235-259 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Materials which are heated by the passage of electricity are usually modeled by a nonlinear coupled system of two partial differential equations. The current equation is elliptic, while the temperature equation is parabolic. These equations are coupled one to another through the conductivities and the Joule effect. A computationally attractive discretization method is analyzed and shown to yield optimal error estimates in H1. © 1993 John Wiley & Sons, Inc.
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    Linearized stability analysis of staggered-grid difference schemes for multidimensional viscous incompressible flows (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993), S. 313-322 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper presents a systematic linearized (frozen coefficient) von Neumann stability analysis for various staggered-grid marker-and-cell (MAC) finite difference schemes for solving viscous incompressible fluid flows. These schemes employ the primitive variables formulation and require the velocity field to be divergent free at every time step. It is illustrated that the stability results for staggered-grid MAC schemes are similar to that for the multidimensional convective-diffusion equation with constant coefficients. © 1993 John Wiley & Sons, Inc.
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    Existence and C0, α regularity of minima of a functional related to the grid-generation problem (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993), S. 261-264 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The existence of C0,α regularity of minima of a functional arising from the grid-generation problem are provided in the case of n-dimensional domains. The functional studied in this article is a linear combination of the length functional, the volume functional, and a higher-order term. The inclusion of the latter guarantees the existence and regularity of a minimum via the direct method for calculus of variations. © 1993 John Wiley & Sons, Inc.
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    Positive solutions and dynamics of a finite difference reaction-diffusion system (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993), S. 285-311 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We investigate the dynamics and methods of computation for some nonlinear finite difference systems that are the discretized equations of a time-dependent and a steady-state reaction-diffusion problem. The formulation of the discrete equations for the time-dependent problem is based on the implicit method for parabolic equations, and the computational algorithm is based on the method of monotone iterations using upper and lower solutions as the initial iterations. The monotone iterative method yields improved upper and lower bounds of the solution in each iteration, and the sequence of iterations converges monotonically to a solution for both the time-dependent and the steady-state problems. An important consequence of this method is that it leads to a bifurcation point that determines the dynamic behavior of the time-dependent problem in relation to the corresponding steady-state problem. This bifurcation point also determines whether the steady-state problem has one or two non-negative solutions, and is explicitly given in terms of the physical parameters of the system and the type of boundary conditions. Numerical results are presented for both the time-dependent and the steady-state problems under various boundary conditions, including a test problem with known analytical solution. These numerical results exhibit the predicted dynamic behavior of the time-dependent solution given by the theoretical analysis. Also discussed are the numerical stability of the computational algorithm and the convergence of the finite difference solution to the corresponding continuous solution of the reaction-diffusion problem. © 1993 John Wiley & Sons, Inc.
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    Three-dimensional singularities of elastic fields near vertices (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993), S. 323-337 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: For the computation of the singular behavior of an elastic field near a three-dimensional vertex subject to displacement boundary conditions we use a boundary integral equation of the first kind whose unknown is the boundary stress. Localization at the vertex and Mellin transformation yield a one-dimensional integral equation on a piecewise circular curve γ in IR3 depending holomorphically on the complex Mellin parameter. The corresponding spectral points and packets of generalized eigenvectors characterize the desired stress field and are computed by a spline-Galerkin method with graded meshes at the corner points of the curve γ. © 1993 John Wiley & Sons, Inc.
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    On least-squares approximations to compressible flow problems (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986), S. 207-228 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A direct finite-element method for computer solutions of compressible potential flow problems is presented. An analysis of least-squares approximation is given, including optimal order estimates for piecewise polynomial approximation spaces. The model problem considered is that of potential flow past a cylinder. Numerical results for the model problem are given for a shock-free subsonic case.
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    An explicit finite difference solution to the convection-dispersion equationPresented at the Fifth IMACS International Symposium on Computer Methods for Partial Differential Equations. (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986), S. 229-237 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: An explicit finite difference equation has been development for the solution of the convection-dispersion equation. This equation has been over the entire range of 2D/vΔx between zero and one, region where no completely satisfactory method has been previously available. No oscillations or numerical dispersion were observed in any of the solutions.
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    Discontinuities across element interfaces in the global element method (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986), S. 259-272 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this article we extend the global element method formalism (Delves and Hall, 1979) for two-dimensional partial differential equations to admit the possibility of solution discontinuities across element interfaces. This extension has particular relevance to problems involving sinks and sources, such as occur in oil well or groundwater management.
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    Efficient boundary element methods for three-dimensional exterior viscous flows (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986), S. 273-297 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: An efficient numerical algorithm for partial differential equations in complicated three-dimensional (3-D) geometries is developed in case of exterior viscous flows. The algorithm essentially consists of a boundary element method, where the resulting algebraic system is solved with a multigrid procedure. Our investigation covers and exact mathematical foundation, a detailed analysis of the discretization error, and some numerical tests with reasonable physical examples.
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    Masthead (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987) 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/num.1690030101
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    A nondispersive and nondissipative numerical method for first-order linear hyperbolic partial differential equations (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987), S. 1-8 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We propose a new numerical method for a solution of first-order linear hyperbolic equations. The leap-frog scheme is converted to a nondispersive scheme by introducing an adjustable constant in a fictitious absorption term. Then the erroneous decrease in th solution is eliminated by solving two equations equivalent to the original equation. The new scheme perfectly preserves the form of a discontinuous solution.
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    Masthead (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987) 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690030201
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    A highly accurate and stable explicit finite difference solution to the convection-dispersion equation with varying velocity and dispersion coefficients (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987), S. 87-99 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Based on the implicit form of the finite difference analogue to the convection-dispersion equation of variable velocity and dispersion coefficients, a highly accurate and stable explicit finite difference scheme has been developed by extending the von Rosenberg linear scheme to the varying cases. This variable velocity and dispersion coefficient scheme has been tested for the entire range of (2D/vΔ x) between zero and unity, the region where no completely satisfactory numerical method has been previously available. No oscillations or numerical dispersion were observed in any of the solutions.
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    Convergence of an element-partitioned subcycling algorithm for the semi-discrete heat equation (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987), S. 131-137 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A convergence analysis is performed for an element-partitioned subcycling algorithm for the semi-discrete heat equation. It is shown that the algorithm generally attains first-order rate-of-convergence.
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    Masthead (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987) 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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    An investigation of chaos in reaction-diffusion equations (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987), S. 139-168 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The purpose of this article is to investigate graphically and numerically the topic of chaos in reaction-diffusion equations. This article is based on the article by Mitchell and Bruch [1]. One- and two-dimensional forms of the reaction-diffusion equation are discretized using the explicit Euler finite difference scheme. Plots are presented to show the effect of bifurcation parameters on the difference equations. Varying these parameters produce single point, periodic, chaotic, intermittent, and divergent solutions.
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    Numerical implementation of the Sinc-Galerkin method for second-order hyperbolic equations (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987), S. 169-185 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A fully Galerkin method in both space and time is developed for the second-order, linear hyperbolic problem. Sinc basis functions are used and error bounds are given which show the exponential convergence rate of the method. The matrices necessary for the formulation of the discrete system are easily assembled. They require no numerical integrations (merely point evaluations) to be filled. The discrete problem is formulated in two different ways and solution techniques for each are described. Consideration of the two formulations is motivated by the computational architecture available. Each has advantages for the appropriate hardware. Numerical results reported show that if 2N + 1 basis functions are used then the exponential convergence rate \documentclass{article}\pagestyle{empty}\begin{document}$ 0\left[{\exp \left({- \kappa \sqrt N} \right)} \right] $\end{document}, κ 〉 0, is attained for both analytic and singular problems.
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    Masthead (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987) 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    High-order methods for elliptic equations with variable coefficientsThis research is supported by the National Science and Engineering Research Council of Canada operating Grants No. 3054 and No. A 5418. (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987), S. 219-227 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this article, we give a simple method for developing finite difference schemes on a uniform square gird. We consider a general, two-dimensional, second-order, partial differential equation with variable coefficients. In the case of a nine-point scheme, we obtain the known results of Young and Dauwalder in a fairly elegant fashion. We show how this can be extended to obtain fourth-order schemes on thirteen points. We derive two such schemes which are attractive because they can be adapted quite easily bnto obtain formulas for gird points near the boundary. In addition to this, these formulas only require nine evaluations for the typical forcing function. Numerical examples are given to demonstrate the performance of one of the fourth-order schemes.
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    Iteration schemes for unknown coefficient problems arising in parabolic equations (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987), S. 313-325 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Additional Material: 6 Ill.
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    An infinite conductivity approximation for eddy current problems (1987)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 3 (1987), S. 327-339 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    Masthead (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988) 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    Superconvergence of the gradient for quadratic triangular finite elements (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 15-32 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
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    Topics: Mathematics
    Additional Material: 6 Ill.
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    An extension of the one-dimensional von Rosenberg finite difference scheme to multidimensional problems of the convection dispersion equation (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 1-13 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The Crank Nicolson implicit form of the finite difference analogue of the convection-dispersion equation does not yield stable solutions when the dispersion coefficient of the medium D is too small compared with the injected fluid velocity V - i.e, 0 〈 (R = 2D/V ΔX) 〈 1. Von Rosenberg developed a simple, highly accurate and stable finite difference scheme for the one-dimensional case of constant velocity and dispersion coefficient for 0 〈 R 〈 1. El-Ageli extended von Rosenberg's method to the multilayered case of varying velocity and dispersion coefficient. This paper goes a step further and extends von Rosenberg's scheme to the linear multidimensional system with constant velocities and dispersion coeffcients when: 0 〈 (Rx, Ry,…,) 〈 1. When this new scheme is compared with the conventional Crank-Nicolson method, it is found that even though there are some restrictions to the spacial grid spacings in relation to the time step, the computational effort is much less and the results obtained are stable.
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    On a moving boundary problem of transitional ballistics (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 69-90 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A hybrid algorithm for calculating the flow around a flat-nosed projectile moving through a fluid is established. At cells of fixed volume away from the projectile, Harten's total variation diminishing (TVD), second-order accurate, shock capturing scheme is utilized. Due to projectile motion, cells adjacent to the projectile can be treated as compressing or expanding in volume. For such cells, a local finite volume approach has been employed to derive a cell update algorithm.Proof of concept for the expanding cell scheme is established through calculating the one-dimensional flow behind a moving piston, whose theoretical solution is well-known. The resulting hybrid scheme is applied to the problem of blast wave simulation in axisymmetric geometries. Flow around a vertical muzzle brake is calculated for the case of an embedded moving projectile.
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    An explicit finite difference scheme based on the modified method of characteristics for solving convection-diffusion problems in one space dimension (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 119-138 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Finite differences are combined with the modified method of characteristics to develop an explicit scheme for solving convection-dominated convection-diffusion problems in one spatial dimension. Error analysis shows that the new algorithm is stable under a mild stability criterion. Problems with known analytical solutions are used to test the algorithm and demonstrate its convergence. Numerical solutions are free of numerical dispersion, undershoot and overshoot. The algorithm is easy to implement and requires small computational times for the test problems considered.
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    A regularization-stabilization technique for nonlinear conservation equation computations (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 165-171 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A regularization procedure for nonlinear conservation equations is introduced and demonstrated to have a stabilizing effect on the numerical solution of the associated approximate problem. Representative results for a least-squares finite-element method are given, and the numerical performance of the stabilization procedure explored. The effect of the regularization term is similar to a local numerical dissipation dependent on the numerical itegration time step.
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    The performance of numerical methods for elliptic problems with mixed boundary conditions (1988)
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    Numerical Methods for Partial Differential Equations 4 (1988), S. 347-361 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider solving linear, second order, elliptic partial differential equations with boundary conditions of types Dirichlet (DIR), mixed (MIX), and nearly Neumann (Neu) by using software modules that implement five numerical methods (one finite element and four finite differences). They represent both the new generation of improved methods and the traditional ones; they are: Hermite collocation plus band Gauss elimination (HC), ordinary finite differences plus band Gauss elimination (5P), ordinary finite differences with Dyaknov iteration (DY), DY with Richardson extrapolation to achieve fourth order convergence (D4), and ordinary finite differences with multigrid iteration (MG). We carry out a performance evaluation in which we measure the grid size and the computer time needed to achieve three significant digits of accuracy in the solution. We compute the changes in these two measures as we change boundary condition types from DIR to MIX and MIX to NEU and then test the following hypotheses: (i) the performance of all the modules is degraded by introducing the derivative terms into the boundary conditions; (ii) finite element collocation (HC) is least affected; (iii) the fourth order modules (HC and D4) are less affected than the other second order modules; and (iv) the traditional 5-point finite differences (5P) are most affected. We establish these hypotheses with high levels of confidence by using several sample problems. The most significant conclusion is that a high order collocation method is preferred for problems with general operators and derivatives in the boundary conditions. We also establish with considerable confidence that these modules have the following rankings in absolute comparative time performance: MG (best), HC and D4, DY, and 5P (worst).
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    On the potential of spectral methods to solve problems in non-Newtonian fluid mechanics (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 35-43 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Spectral techniques for solving problems in non-Newtonian fluid mechanics are introduced. Following the work of Coleman (J. Non-Newtonian Fluid Mech.; 15, 227-238 [1984]), the governing equations for the creeping flow of a co-rotational Maxwell fluid are written in terms of the Airy stress function and a stream function. This ensures that the continuity and momentum equations are automatically satisfied. The choice of trial functions for solving a one-dimensional model problem using spectral methods is discussed. Methods for treating unbounded domains and accurately representing reentrant boundary singularities within the spectral context are also considered.
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    A numerical study of the derivatives of solutions of the wave equation with a singular forcing term at quenching (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 53-75 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Additional Material: 9 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690050106
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    An experimental performance analysis for the rate of convergence of collocation on general domains (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 45-52 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper presents an experimental performance analysis for the rate of convergence of collocation on general domains using a bicubic Hermite basis. Twenty domains are selected for the experiment from the population of PDEs on nonrectangular domains found in Realistic PDE Solutions for Non-rectangular Domains (C. J. Ribbens and J. R. Rice, CSD-TR 639, Department of Computer Sciences, Purdue University, 1986), including one rectangle for comparison. The result shows that the convergence of the ELLPACK module COLLOCATION behaves as O(h4) on all 19 of the nonrectangular domains. This set includes a large variety of nonrectangular domains (only two have reentrant corners). We conclude that, with very high probability, this collocation module has O(h4) convergence on general domains.The experiment is made by using the Performance Evaluation System (PES) of ELLPACK, which includes the population of PDEs on nonrectangular domains. Several performance analysis tools are used to analyze the rate of convergence, the most informative is a visual examination of the convergence behavior.
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    Asymptotic stability of the numerical solution for an integrodifferential reaction-diffusion systemWork performed in the context of the special program “Infections Diseases” of the Centro Nazionale Delle Ricerche. (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 79-86 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper presents the study of the numerical solution of a reaction-diffusion system involving a reaction term of integral type arising from biological models. By means of a monotone approach we introduce upper and lower solutions and then we show the existence and the asymptotic behavior of nonnegative numerical solutions. To this end, we require the positivity of the numerical scheme and so we can use some properties of positive and M-matrices. Finally we give some sufficient conditions to verify the asymptotic stability of the numerical solution.
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    Preconditioned iterative algorithms for large sparse unsymmetric problems (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 107-120 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A numerical study of the efficiency of the generalized conjugate residual methods (GCR) is performed using three different preconditioners all based upon an incomplete LU factorization. The GCR behavior is evaluated in connection with the solution of large, sparse unsymmetric systems of equations, arising from the finite element integration of the diffusion-convection equation for 2-dimensional (2-D) and 3-D problems with different Peclet and Courant numbers. The order of the test matrices ranges from 450 to 1700. Results from a set of numerical experiments are presented and comparisons with preconditioned GCR methods and with direct method are carried out.
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    On the accuracy and convergence of conjugate flux approximations (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 143-156 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We prove that the L2-projections of derivatives of piecewise bilinear or linear finite element approximations of smooth solutions of elliptic boundary-value problems on the interior of uniform meshes, converge in L2 and L∞ at a rate faster than that of derivatives of the approximations themselves.
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    A finite difference approach to the equations of age-dependent population dynamics (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 157-168 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We establish the convergence of the finite difference scheme for the nonlinear equations of population dynamics proposed by Guertin and MacCamy. The applicability of the discrete equations to establish qualitative properties of the solution to the continuous problem is also illustrated.
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    Errata (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. i 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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    Masthead (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989) 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690050301
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    A boundary integral approach to Stefan problems with many phase changes (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 173-183 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper considers a boundary integral approach to Stefan problems that have multiple phase changes and satisfy the Laplace equation in each phase. It is shown that, by introducing artificial phase changes, the effects of moderate diffusivity can be incorporated. The paper includes the results from several computer simulations.
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    A new numerical approach for the advective-diffusive transport equation (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 203-226 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A new numerical solution procedure is presented for the one-dimensional, transient advective-diffusive transport equation. The new method applies Herrera's algebraic theory of numerical methods to the spatial derivatives to produce a semi-discrete approximation. The semi-discrete system is then solved by standard time marching algorithms. The algebraic theory, which involves careful choice of test functions in a weak form statement of the problem, leads to a numerical approximation that inherently accommodates different degrees of advection domination. Algorithms are presented that provide either nodal values of the unknown function or nodal values of both the function and its spatial derivative. Numerical solution of several test problems demonstrates the behavior of the method.
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    Recovery of an unknown specific heat by means of overposed data (1990)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 6 (1990), S. 1-16 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we consider a class of inverse problems in which an unknown function, c(.), is to be determined from a parabolic initial-value problem, with overposed Dirichlet data along a portion of the boundary. A mapping between the overposed data and the unknown coefficient is obtained in the form of a singular integral equation. This is solved by iteration, and the resulting fixed point is shown to be the solution of the inverse problem. Sufficient conditions for convergence of this method, as well as an extension to the case of an unknown thermal conductivity, are given.
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    Finite elements for transonic potential flows (1990)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 6 (1990), S. 17-42 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A finite element method is used for the computation of entropy solutions to the transonic full potential equation. Physically correct solutions with sharp and correctly placed shocks were obtained. (AMS (MOS) 1980 Mathematics subject classifications: 65N30, 76N15, 35M05, 76H05, 49D10, 35A40, 35L67.)
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    From the editors (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. i 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    A numerical study of chaos in a reaction-diffusion equation (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 13-23 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A study is made using numerical experiments to see the effect of the parameters in the explicit Euler-discretized form of a one-dimensional, nonlinear, reaction-diffusion equation. Based on a series of these experiments, one of the main results obtained is that diffusion, which is usually perceived as having a stabilizing effect, is able to produce chaotic as well as divergent numerical solutions. Furthermore, the discretization parameters are also able to produce chaotic results. From the results presented herein, it is shown that varying the parameters can produce solutions that are single numbers, periodic, aperiodic (chaos), or divergent.
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    Masthead (1990)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 6 (1990) 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/num.1690060201
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    An analysis of the finite element method for natural convection problems (1990)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 6 (1990), S. 115-126 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We derive stability properties and error estimates for the finite element method when used to approximate heat flow in a fluid enclosed by a solid medium. The coupled Navier Stokes system involves the Boussinesq equations in the fluid-filled cavity linked through an interface with heat conduction in the solid enclosing the fluid. As we assume no extra regularity then can be shown to hold under mild restriction on the data (at least over a small time interval in R3), we focus primarily on low order finite element spaces.
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    Pointwise superconvergence of recovered gradients for piecewise linear finite element approximations to problems of planar linear elasticity (1990)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 6 (1990), S. 59-74 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we consider piecewise linear finite element approximations to the problem of planar linear elasticity, and present some new estimates for the pointwise (L∞) superconvergence of a recovered gradient function to the gradient of the true solution. This extends to linear elasticity the previous work of the present and other authors on L∞ results for Poisson problems, and at the same time, to the L∞ norm the previous L2 results of the authors for linear elasticity.
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    Path integral numerical methods for evolution equations (1990)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 6 (1990), S. 127-135 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A universal method for representing solutions of the initial value problem ut = -2πiDu is presented for a large class of pseudo-differential operators. The representations take the form of a path integral-like operator. Three numerical approaches for evaluating these operators are treated. Practical interest in such universal methods depends on massively parallel computing systems.
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    Convergence properties of a class of boundary element approximations to linear diffusion problems with localized nonlinear reactions (1990)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 6 (1990), S. 75-108 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider a boundary element (BE) Algorithm for solving linear diffusion desorption problems with localized nonlinear reactions. The proposed BE algorithm provides an elegant representation of the effect of localized nonlinear reactions, which enables the effects of arbitrarily oriented defect structures to be incorporated into BE models without having to perform severe mesh deformations.We propose a one-step recursion procedure to advance the BE solution of linear diffusion localized nonlinear reaction problems and investigate its convergence properties. The separation of the linear and nonlinear effects by the boundary integral formulation enables us to consider the convergence properties of approximations to the linear terms and nonlinear terms of the boundary integral equation separately.For the linear terms we investigate how the degree of piecewise polynomial collocation in space and the size of the spatial mesh relative to the time step affects the accumulation of errors in the one-step recursion scheme. We develop a novel convergence analysis that combines asymptotic methods with Lax's Equivalence Theorem. We identify a dimensionless meshing parameter θ whose magnitudé governs the performance of the one-step BE schemes. In particular, we show that piecewise constant (PWC) and piecewise linear (PWL) BE schemes are conditionally convergent, have lower asymptotic bounds placed on the size of time steps, and which display excess numerical diffusion when small time steps are used. There is no asymptotic bound on how large the tie steps can be-this allows the solution to be advanced in fewer, larger time steps. The piecewise quadratic (PWQ) BE scheme is shown to be unconditionally convergent; there is no asymptotic restriction on the relative sizes of the time and spatial meshing and no numerical diffusion. We verify the theoretical convergence properties in numerical examples. This analysis provides useful information about the appropriate degree of spatial piecewise polynomial and the meshing strategy for a given problem.For the nonlinear terms we investigate the convergence of an explicit algorithm to advance the solution at an active site forward in time by means of Caratheodory iteration combined with piecewise linear interpolation. We consider a model problem comprising a singular nonlinear Volterra equation that represents the effect of the term in the BE formulation that is due to a single defect. We prove the convergence of the piecewise linear Caratheodory iteration algorithm to a solution of the model problem for as long as such a solution can be shown to exist. This analysis provides a theoretical justification for the use of piecewise linear Caratheodory iterates for advancing the effects of localized reactions.
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    Finite elements in h(curl) with applications to elasticity (1990)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 6 (1990), S. 167-175 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/num.1690060206
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    A generalized finite difference method for solid mechanics (1990)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 6 (1990), S. 137-152 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Procedures are developed that improve the applicability of the finite difference method to problems in solid mechanics. This is accomplished by formulating the coefficients of the Taylor series expansion used to approximate derivative quantities in terms of physically interpretable strain gradients. Improvements realized include modeling of boundary conditions that has intuitive appeal and the use of irregular grids in a natural manner. These developments are demonstrated for the analysis of plane stress problems with traction boundary conditions. The results compare well with finite element solutions. The approach suggests further generalization of the finite difference method.
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    Masthead (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985) 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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    Principles of electricity and economics in fluid dynamics (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 145-157 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present essential connections between electrical field decomposition and network theory, economic decentralization theory, and the work of Kron on decomposition of systems for computational fluid dynamics.
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    Finite-element stream function solutions of transonic rotational internal and external flows (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 127-144 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The finite-element method is applied to the stream function formulation of transonic flows. Numerical dissipation, necessary for the calculation of mixed flows with shocks, is introduced via the artificial compressibility method. The classical problem of double-valuedness of the mass flux versus Mach number is resolved by direct integration of the vorticity equation. Solutions are obtained for isolated airfoils, the blade-to-blade cascade equation, as well as the radial equilibrium equation governing the hub-to-shroud through flow in turbomachinery.
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    Unified approach to numerical methods, part II: Finite elements, boundary methods, and their coupling (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 159-186 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This is the second in a series of three papers devoted to the presentation of a direct procedure of analysis of numerical methods for partial differential equations. The procedure consists of applying the method of weighted residuals and then interpreting the resulting equations by means of Green's formulas for discontinuous functions. Here, the general Green's formulas for operators defined in discontinuous fields developed in the first article, are applied to formulate the method of weighted residuals for arbitrary linear operators. Finite elements, boundary methods, and general procedures for coupling finite elements and boundary methods are discussed.
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    Mixed finite element methods for computing groundwater velocities (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 195-207 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Central to the understanding of problems in water quality and quantity for effective management of water resources is the development of accurate numerical models to stimulate groundwater flows and contaminant transfer. We discuss several important difficulties arising in modeling of subsurface flow and present promising numerical procedures for alleviating these problems. Furthermore, we describe mixed-finite element techniques for accurately approximating fluid velocities, and review computational results on a variety of hydrologic problems.
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    Monotonicity and finite element methods for hyperbolic initial value problems (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 187-194 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A well known theórem about super- and subfunctions for the solution of hyperbolic initial value problems constructs differentiable functions as upper and lower bounds (see Walter [1], 21 XIII). The proof can be done by transforming the differential equation problem into a set of integral equations, using the monotonicity-properties of the arising integral operators. This proof needs an integral representation for twice differentiable functions. It is shown that this proceeding can be generalized to get upper and lower bounds in terms of finite element functions. To do this, we give an integral representation for continuous, piecewise differentiable functions, including the discontinuities of their derivatives. Then the generalization of the classical proof yields interface conditions for the finite element functions. Finally, it is demonstrated how to realize numerically these conditions.
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    URL: http://dx.doi.org/10.1002/num.1690010303
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    A compact finite element method for elastic bodies (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 209-228 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A nonconforming finite element method is described for treating linear equilibrium problems, and a convergence proof showing second order accuracy is given. The close relationship to a related compact finite difference scheme due to Phillips and Rose [1] is examined. A Condensation technique is shown to preserve the compactness property and suggests an approach to a certain type of homogenization.
    Additional Material: 7 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690010305
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    Masthead (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985) 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690010401
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    A finite element collocation method for variably saturated flows in porous media (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 229-239 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: One common formulation of Richard's equation for variably saturated flows in porous media treats pressure head as the principal unknown and moisture content as a constitutive variable. Numerical approximations to this “head-based” formulation often exhibit mass-balance errors arising from inaccuracies in the temporal discretization. This article presents a finite-element collocation scheme using a mass-conserving formulation. The article also proposes a computable index of global mass balance.
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    Unified approach to numerical methods. III. Finite differences and ordinary differential equations (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 241-258 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This is the third of a sequence of articles devoted to the presentation of a direct method of analysis recently developed by one of the authors. The approach is quite general, since it is applicable to any linear operator, symmetric or nonsymmetric, regardless of its type. In particular, the theory includes steady-state and time-dependent problems. In this article the method is applied to ordinary differential equations, which constitute a very convenient illustration of its application because the analysis can be carried out in an exhaustive manner. General algorithms which yield the exact values of the solution or its derivatives are obtained in this manner.
    Additional Material: 4 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690010402
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    Variable trial functions and the CVBEM (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 259-278 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The Complex Variable Boundary Element Method; or CVBEM will be developed with respect to a variable trial function definition over each boundary element. The benefits in using this technique are that the modeling error in matching the prescribed boundary conditions (there is no error in satisfying the Laplace equation) is reduced without the addition of nodal points to the problem discretization. Consequently, the n × n matrix requirements are not increased when using this new approach.
    Additional Material: 21 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690010403
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    A network model for two-fluid, two-phase flow (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 295-313 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A systematic derivation is given of two-fluid conservation laws on an arbitrary network. These laws are shown to conserve the mass and total energy of the mixture in the network. Numerical simulations are presented to illustrate their use.
    Additional Material: 7 Ill.
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    Erratum (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 329-329 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690010407
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    Two-phase laminar axisymmetric jet flow: Explicit, implicit, and split-operator approximations (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 279-294 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: An hybrid Eulerian-Lagrangian numerical scheme is developed for a two-phase problem and four finite-difference schemes are compared. For this purpose, the problem of hydrodynamic and thermal interactions between a fuel spray and a mixing region of two laminar, unconfined axisymmetric jets is formulated in terms of a set of parabolic differential equations for the gas phase and a set of Lagrangian ordinary differential equations for the condensed phase. Consistent, second-order accurate hybrid numerical schemes, with the exception of the explicit scheme with an accuracy between linear and quadratic, are used to solve these equations. The subset of gas-phase equations has been solved by four different numerical methods: a predictor-corrector explicit method, a sequential implicit method, a block implicit method, and a symmetric operator-splitting method. The subsystem of liquid-phase equations is solved along the droplet trajectories by a second-order Runge-Kutta scheme. The computations have been made to predict the hydro-dynamic and thermal mixing regions of the gas phase as well as the trajectories of each individual group of droplets. In addition, the size, velocity and temperature associated with each group are predicted along these trajectories. The relative merits of the above four difference-schemes are discussed by constructing effectiveness curves. At low error tolerances, the sequential implicit method gives the best results, where for large error tolerances, the explicit and operator splitting give better results. The block implicit scheme is the least effective at all accuracy requirements.
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    On operator splitting of the Euler equations consistent with Harten's second-order accurate TVD scheme (1985)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 1 (1985), S. 315-327 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Operator splitting is necessary in order to apply Harten's second-order accurate, total-variation-diminishing (TVD) method for shock capturing to higher-dimensional problems. Here two such splittings of the Euler equations are considered. By analytical means, one splitting is shown to be inconsistent; the other, scheme satisfies a fundamental necessary condition for consistency. Numerical results from a typical blast wave calculation employing this scheme are exhibited.
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    Masthead (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986) 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690020101
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    Nonreflecting upwind boundaries for hyperbolic equations (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986), S. 1-12 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: It is commonly, but erroneously, assumed that the best way to treat upwind (closed) boundaries in numerical approximations of hyperbolic equations consists in a literal transcription, letting the numerical value be equal to the prescribed value. This results in a total reflection of spurious solutions that may arrive at the boundary from the computing domain. Those reflected solutions cannot be distinguished from consistent solutions, and they may seriously degrade the overall accuracy. We show that modifications of this treatment of the boundary may result in the absorption of spurious solutions. The effect of the absorbing properties of these boundary schemes is analyzed in Fourier space. We also analyze their numerical stability properties, and their effect on the accuracy of solutions generated in response to a time dependent boundary condition.
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    A family of numerical methods for diffusion and reaction-diffusion equations (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986), S. 31-45 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A family of methods is developed for the numerical solution of second-order parabolic partial differential equations in one space dimension. The methods are second-, third-, or fourth-order accurate in time; five of them are seen to be L0-stable in the sense of Gourlay and Morris, while the sixth is seen to be A0-stable, The methods are tested on four problems from the literature, three diffusion problems and one reaction-diffusion problem.
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    URL: http://dx.doi.org/10.1002/num.1690020104
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    Stability of penalty finite-element methods for nonconforming problems (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986), S. 13-29 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Penalty methods have been proposed as a viable method for enforcing interelement continuity constraints on nonconforming elements. Particularly for fourth-order problems in which C1-continuity leads to elements of high degree or complex composite elements, the use of penalty methods to enforce the C1-continuity constraint appears promising. In this study we demonstrate equivalence of the finite-element penalty method to a hybrid method and provide a stability analysis which implies that the penalty method is stable only if reduced integration of a certain order is used. Numerical experiments confirm that the penalty method fails if this condition is not met.
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    The effects of symmetric/asymmetric boundary conditions on the flow of an internally heated fluid (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986), S. 47-61 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The two-dimensional flow of an internally heated fluid in a circular vessel has been investigated by using a finite-control-volume numerical model. It has been found that, when symmetry constraints are imposed along the vertical midplane of the vessel, two distinct flow patterns can he predicted for the same operating conditions, a jet-momentum-dominated pattern and buoyancy-dominated pattern. These patterns occur in an operating region where momentum and buoyancy forces are of comparable magnitude. However, when the symmetry constraints are removed and the full vessel cross section is modeled, only the buoyancy-dominated pattern is observed. Results for the two cases are described and possible reasons for the differences in behavior are discussed.
    Additional Material: 16 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690020105
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    Masthead (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986) 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690020201
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    Fourth-order accurate one-step integration methods with large imaginary stability limits (1986)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 2 (1986), S. 63-70 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: One-step integration methods of fourth-order accuracy using an odd number of function evaluations K, to solve dy/dt = A · y, are proposed. These methods have an imaginary stability limit \documentclass{article}\pagestyle{empty}\begin{document}$ S_{1\;} = \sqrt {(K - 1)^2 - 4} $\end{document}. In the case K = 5 the Kutta-Merson method results.
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    URL: http://dx.doi.org/10.1002/num.1690020106
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