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  • 1
    Electronic Resource
    Electronic Resource
    Masthead (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988) 
    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.1690040101
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  • 2
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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 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040103
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  • 3
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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 
    Details
    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.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040102
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  • 4
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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 
    Details
    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.
    Additional Material: 27 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040106
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  • 5
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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 
    Details
    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.
    Additional Material: 8 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040203
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  • 6
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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 
    Details
    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.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040302
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  • 7
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    The performance of numerical methods for elliptic problems with mixed boundary conditions (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 347-361 
    Details
    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).
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040407
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  • 8
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    A penalty finite element model for axisymmetric flows of non-Newtonian fluids (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 33-56 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Additional Material: 12 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040104
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  • 9
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    On the finite element approximation of incompressible flows of an electrically conducting fluidThis work was performed under the auspices of the U. S. Department of Energy and under Grant AF-AFOSR-84-0131 while the author was in residence at the University of Pittsburgh. (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 57-68 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider the finite element approximation of incompressible flows field of an electrically conducing fluid in the presence of a magnetic where it is assumed that this field is prescribed. A weak form is chosen that is similar in some respects to a weak form used by many authors for the Navier-Stokes equations. Existence and uniqueness results are presented for the weak problem. A finite element Algorithm is given for the approximate solution of the weak problem and error estimates are derived.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040105
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  • 10
    Electronic Resource
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    Masthead (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988) 
    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.1690040201
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  • 11
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    Optimization of computational grids (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 95-117 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider the problem of mesh grading or gridr-point redistribution using optimization techniques. Some representate forms of the objective functions are considered and the assocated scaling parameterization and nonlinear programming algorithms examined. Numerical results for representative test problems involving two-and three-dimensional grids are presented.
    Additional Material: 8 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690040202
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  • 12
    Electronic Resource
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    Masthead (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988) 
    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.1690040301
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  • 13
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    Numerical comparison of preconditionings for large sparse finite element problems (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 139-157 
    Details
    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 modified conjugate gradients (MCG) is performed using different preconditioning schemes. The MCG behavior is evaluated in connection with the solution of large linear sets of symmetric positive definite (p.d.) equations, arising from the finite element (f.e.) integration of partial differential equations of parabolic and elliptic type and the analysis of the leftmost eingenspectrum of the corresponding matrices. A simple incomplete Cholesky factorization ICCG(O) having the same sparsity pattern as the original problem is compared with a more complex technique ICAJ (Ψ) where the triangular factor is allowed to progressively fill in depending on a rejection parameter Ψ. The performance of the preconditioning algorithms is explored on finite element equations whose size N ranges between 150 and 2300. The results show that an optimal Ψopt may be found which minimizes the overall CPU time for the solution of both the linear system and the eigenproblem. The comparison indicates that ICAJ (Ψopt) is not significantly more efficient than ICCG(O), which therefore appears to be a simple, robust, and reliable method for the preconditioning of large sparse finite element models.
    Additional Material: 12 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040204
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  • 14
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    Six combinations of the Ritz-Galerkin and finite element methods for elliptic boundary value problems (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 197-218 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Overall comparisons are made for six efficient combinations of the Ritz-Galerkin and finite element methods for solving elliptic boundary value problems with singularities or interfaces. The comparisons are done by using theoretical analysis and numerical experiments. Significant relations among the six combinations are also found. A survey of the six combinations and their coupling strategies are given. These combinations are important not only for matching the Ritz-Galerkin method and the finite element method but also for matching other numerical methods such as the Ritz-Galerkin method and the finite difference method.
    Additional Material: 7 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040304
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  • 15
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    Numerical solution of the Fokker Planck equation using moving finite elements (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 219-232 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Numerical solution of the Fokker Planck equation for the probability density function of a stochastic process by traditional finite difference or finite element methods produces erroneous oscillations and negative values whenever the drift is large compared to the diffusion. Upwinding schemes to eliminate the oscillations introduce false numerical diffusion because it is impossible to make the one step drift large enough to match the original equation without making the one step diffusion too large. A variation of the moving finite element method is presented that overcomes these difficulties by using basis functions that satisfy the drift part of the equation by moving along the trajectories of the deterministic dynamical system associated with the stochastic process. A Galerkin type method can then be used to find the coefficients in the remaining pure diffusion equation. Solutions of two test equations are presented to illustrate the effectiveness of the method.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040305
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  • 16
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    Masthead (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988) 
    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.1690040401
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  • 17
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    Solution of the advection-dispersion equation with free exit boundary (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 301-313 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: I investigated the exit boundary condition for the advection-dispersion equation and found that in numerical solutions of this equation, using Galekin finite elements, a free exit boundary condition requiring no a priori information is possible, provided the advective component in the numerical equations is of sufficient magnitude relative to the dispersive component. Since the relationship between these two components is controlled by the spatial discretization through the grid Peclet number, the free exit boundary condition can in fact be applied whenever there is a non-zero advective component. The numerical solution in a finite domain with free exit boundary, using a correctly proportioned spatial discrezation, behaves like an infinite-domain solution.
    Additional Material: 4 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690040403
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  • 18
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    A Tensor product generalized ADI method for the method of planes (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 283-300 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider solving separable, second order, linear elliptic prtial differential equations in three independent variables. If the partial differential opertor separates into two terms, one depending on x and y, and one depending on z, then we use the method of planes to obtain a discrete problem, which we write in tensor product from as \documentclass{article}\pagestyle{empty}\begin{document}$$ \left({T_z \otimes B_{xy} + I \otimes A_{xy} } \right)C = F $$\end{document} We apply a new interative method, the tensor product generalized alternating direction implicit method, to solve the discrete problem. We study a specific implementation that uses Hermite bicubic collocation in the xy direction and symmetric finite differences in the z direction. We demostrate that this method is a fast and accurate way to solve the large linear systems arising from three-dimensional elliptic problems.
    Additional Material: 5 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690040402
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  • 19
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    Numerical methods for structured population models: The Escalator Boxcar Train (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 173-195 
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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 integration method is introduced for the class of hyperbolic partial differential equations that occur in physiologically structured population models. These equations describe the time evolution of the population density-function over the individual state-space Ω. Exploiting the biological interpretation of the equation, this density-function is represented by a set of moments over a collection of subdomains in Ω, moving along the characteristics of the partial differential equation (PDE). These moments are readily interpreted as numbers of individuals in a cohort, mean individual state in a cohort, etc. The numerical method consists of approximating the differential equations.The method appears to be very efficient for this special type of PDE. It combines such desirable properties as as lack of dispersion and dissipation and a preservation of monotonicity of the density function along the characteristics with a strong relation to the biological background of the equations for these moments to arrive at a closed system of ordinary differential equations. The method also allows one to incorporate the usual type of nonlinearities that occur in physiologically structured population models. A numerical example is presented as an illustration of the application of the method.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690040303
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  • 20
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    On the numerical analysis of analytical nodal methods (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 233-254 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This article presents the basic numerical analysis of the analytical nodal methods, which were originally developed in the late 1970s in relation with static and dynamic nuclear reactor calculations but are actually applicable to the numerical solution of partial differential equations (PDEs) in general, over fairly regular domains. The basic idea consists in “transverse integrating” the original PDE over all the variables minus one, leading to sets of 1D equations which are then solved in an “analytical” way, using fundamentals as well as particular solutions of the corresponding 1D operators. After examining the existing analytical methods in a critical way, we propose a more satisfactory extended analytical formalism. Superconvergence results finally lead us to useful conclusions with respect to the choice of a particular scheme.
    Additional Material: 3 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690040306
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  • 21
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    Two different approaches for the treatment of boundary singularities (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 255-282 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The finite element treatment of boundary singularities in elliptic problems has demanded that special techniques be developed. Many of these use some form of singular element in the neighborhood of a singular point.For a homogenous second-order problem, defined in a domain with a polygonal boundary, we studied three cases with different singularity orders. Some results about the accuracy of the solution are presented. Numerical results have been obtained using Akin singular triangular elements with three and six nodes and quadrilateral elements having four and eight nodes. The behavior of the gradient in each of these elements is also analyzed. However, these elements are not completely satisfactory, so that an alternative technique using curved isoparametric elements is given here. The results obtained with the two methods are compared. Conclusions about numerical accuracy of each method, the order of integration and the simplicity of application are made.
    Additional Material: 28 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690040307
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  • 22
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    Downstream implicit schemes for advection (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 315-318 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Downstream-weighted implicit difference schemes for pure advection are stable for Courant numbers λ ≥ 1. They are numerically diffusive when λ 〉 1.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/num.1690040404
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  • 23
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    Stability improvement of the Euler forward method for initial value problems (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 319-327 
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    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The Euler forward method is transformed into a highly stable two-step explicit algorithm by use of a relaxation paramenter α. The stability of the new explicit family can be very high and linearly proportional to the absolute value of α. An adaptive stability approach with a matrix of parameters α is used that strongly increases the effeciency of the method. The one-dimensional finite element solutions for linear and nonlinear transient heat conduction problems show the stability, accuracy, and computational efficiency characteristics of the proposed algorithms.
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    URL: http://dx.doi.org/10.1002/num.1690040405
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  • 24
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    A finite element method for a two-sex model of population dynamics (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 329-345 
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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 scheme is described to approximate the solution of a nonlinear and non-local system of integro-differential equations that models the dynamics of a two-sex population. Crank-Nicolson time discretization is used and error estimates are derived for the appoximation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040406
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