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  • Articles  (194)
  • Numerical Methods  (194)
  • 1990-1994  (194)
  • 1950-1954
  • Mathematics  (194)
  • 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
    Electronic Resource
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690070206
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  • 4
    Electronic Resource
    Electronic Resource
    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
    Electronic Resource
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690070203
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  • 6
    Electronic Resource
    Electronic Resource
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690070205
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  • 7
    Electronic Resource
    Electronic Resource
    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
    Electronic Resource
    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
    Electronic Resource
    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
    Electronic Resource
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690070306
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