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SIAM J. on Scientific Computing

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1982

Volume 3, Issue 4, pp. 387-515


Theoretical and Practical Aspects of a Multigrid Method

P. Wesseling

SIAM J. Sci. and Stat. Comput. 3, pp. 387-407 (21 pages) | Cited 20 times

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A multigrid method is described. A novel item is the use of incomplete $LU$ decomposition for smoothing. Numerical experiments show that its speed and robustness compare favorably with other multigrid methods. A fairly simple rate of convergence proof is presented.

Stable Boundary Approximations for Implicit Time Discretizations for Gas Dynamics

Bertil Gustafsson and Joseph Oliger

SIAM J. Sci. and Stat. Comput. 3, pp. 408-421 (14 pages) | Cited 3 times

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We consider the problem of constructing stable difference methods for the initial boundary value problem for the linearized equations of gas dynamics in one space dimension using the implicit time differencing methods considered by Beam and Warming [2]. Centered spacial differences are used in the interior. We investigate the stability of this class with two forms of extrapolation for the scalar outflow problem. (We consider the problem of specifying data in the primitive variables and computing in terms of the conservative variables in the interior.) We show that the whole class of methods is stable for the subsonic inflow and outflow problems with various data specifications and extrapolation methods. We also show that the methods considered are stable for the solid wall boundary problem when we set $u = 0$and use one-sided differences in the other equations.

Why Particle Methods Work

J. J. Monaghan

SIAM J. Sci. and Stat. Comput. 3, pp. 422-433 (12 pages) | Cited 83 times

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The theme of this paper is that particle methods are closely related to both finite difference and spectral methods because the three methods can be considered special cases of interpolation by kernel estimation. The kernels for a number of special cases are given in detail, and the accuracy of the resulting interpolation is analyzed. A general procedure for deriving equations for numerical work from the equations of hydrodynamics is described. It is applied to the derivation of the SPH equations which conserve linear and angular momentum exactly.

A Method for Computing the Integral of the Bivariate Normal Distribution Over an Arbitrary Polygon

A. R. DiDonato and R. K. Hageman

SIAM J. Sci. and Stat. Comput. 3, pp. 434-446 (13 pages) | Cited 2 times

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An efficient programmable procedure is given for evaluating the integral of the bivariate normal distribution (IBND) over an arbitrary polygon $\Pi $. The class of arbitrary polygons includes the subclasses: simple polygons, limit elements of sequences of uniformly bounded $N$-sided simple polygons with the same orientation, and self-intersecting (SI) polygons. For a given element $\Pi $ defined by $N$ ordered points in the plane, the subclass need not be specified. The method evaluates the IBND over $N$ exterior angular regions $A_1 ,A_2 , \cdots ,A_N $ of $\Pi $ to determine the IBND for $\Pi $. If $\Pi $ is SI, a quantity called the “winding number” of $\Pi $ is introduced which is given by the sum of the angular measures of the $A_i (i = 1,2 \cdots ,N)$ divided by $2\pi $. A detailed numerical example, using a Fortran IV program, with approximately 9-decimal-digit accuracy is included.

On the Solution of the Finite Element Equations for Nonlinear Shell Analysis

Lois Mansfield

SIAM J. Sci. and Stat. Comput. 3, pp. 447-459 (13 pages) | Cited 1 time

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The relative efficiencies of the finite element methods derived from the potential energy formulation and from the mixed formulation for nonlinear shell analysis are compared. The result of this comparison is that the mixed method is considerably more efficient.

Discrete Weighted Mean Approximation of a Model Convection-Diffusion Equation

E. C. Gartland, Jr.

SIAM J. Sci. and Stat. Comput. 3, pp. 460-472 (13 pages) | Cited 8 times

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Five-point finite-difference approximations are considered for a model (linear, constant-coefficient) convection-diffusion equation in two dimensions. Standard difference schemes for such problems behave badly when the convective terms are dominant. A new discretization is derived from a local integral representation of the true solution. This derivation is analogous to the way that the discrete Laplacian can be derived from the mean-value property of harmonic functions, and it generalizes an approach due to Allen and Southwell [Quart. J. Mech. Appl. Math., 8 (1955), pp. 129–45]. Also discussed is how the strong upwind bias of this and other discretizations serves to make more stable some methods of the two-sweep or marching type for the direct solution of the resulting linear algebraic equations.

Preconditioning and Coarse Grid Corrections in the Solution of the Initial Value Problem for Nonlinear Partial Differential Equations

P. J. van der Houwen and H. B. de Vries

SIAM J. Sci. and Stat. Comput. 3, pp. 473-485 (13 pages) | Cited 2 times

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The numerical solution of nonlinear, time-dependent partial differential equations is discussed. An initial value problem for a system of ODE’S is obtained by the method of lines, and an implicit linear multistep method is applied to this initial value problem. Using Newton type iteration the nonlinear implicit relations are replaced by a sequence of linear equations. The linear problems are preconditioned by applying incomplete $LU$-decomposition and are then solved by iterative refinement. The convergence is accelerated by introducing coarse grid corrections. Numerical examples are given and a comparison is made with other integration techniques.

Comparison of Two Algorithms for Solving Large Linear Systems

Zahari Zlatev, Jerzy Wasniewski, and Kjeld Schaumburg

SIAM J. Sci. and Stat. Comput. 3, pp. 486-501 (16 pages) | Cited 4 times

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Assume that the coefficient matrix $A$ in the system $Ax = b$ is large and sparse. Consider the following two algorithms: DS (where the system is solved by a direct use of Gaussian elimination) and IR (where the use of Gaussian elimination is combined with the use of a large drop tolerance and followed by iterative refinement). Assume that some sparse technique is implemented with both DS and IR. The performance of two codes, the NAG subroutines (which are based on DS) and the RECKU subroutines (which are based on IR) are compared on a wide set of test matrices. The comparison shows that the second algorithm, IR, performs better in general. The computing time and/or the storage needed may be reduced considerably when the IR algorithm is used. Moreover, this algorithm normally provides a reliable estimate of the accuracy of the computed solution. When the problems are time and storage consuming, IR is much better (it gives a reduction in the computing time of up to 10 times and a reduction in the storage of up to 2–3 times). It is shown that IR is very efficient when linear least-squares problems are solved by the use of augmented matrices.

The Simulation of Generalized Inverse Gaussian and Hyperbolic Random Variables

A. C. Atkinson

SIAM J. Sci. and Stat. Comput. 3, pp. 502-515 (14 pages) | Cited 9 times

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Computer algorithms are described for simulation of the generalized inverse Gaussian, generalized hyperbolic and hyperbolic distributions. The efficiencies of the algorithms are found. Timing comparisons with the best available algorithms for sampling the gamma distribution show the new algorithms to be acceptably fast. The extension to sampling multivariate generalized hyperbolic distributions is escribed. Listings of Fortran implementations of the algorithms are available.
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