Higher order compact implicit schemes for the wave equation

Authors:
Melvyn Ciment and Stephen H. Leventhal

Journal:
Math. Comp. **29** (1975), 985-994

MSC:
Primary 65M05; Secondary 35L05

DOI:
https://doi.org/10.1090/S0025-5718-1975-0416049-2

MathSciNet review:
0416049

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Abstract | References | Similar Articles | Additional Information

Abstract: Higher order finite-difference techniques have associated large star systems which engender complications near the boundary. In the numerical solution of hyperbolic equations, such boundary conditions require careful treatment since errors or instabilities generated there will in general pollute the entire calculation. To circumvent this difficulty, we use a class of implicit schemes suggested by H.-O. Kreiss, which achieves the highest order of accuracy possible on the smallest (most compact) mesh system. Here we develop a scheme which approximates the wave equation,

*L*-shaped domains.

**[1]**G. Fairweather and A. R. Mitchell,*A high accuracy alternating direction method for the wave equation*, J. Inst. Math. Appl.**1**(1965), 309–316. MR**0210331****[2]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[3]**Milton Lees,*Alternating direction methods for hyperbolic differential equations*, J. Soc. Indust. Appl. Math.**10**(1962), 610–616. MR**0172475****[4]**S. McKEE, "High accuracy A.D.I. methods for hyperbolic equations with variable coefficients,"*J. Inst. Math. Appl.*, v. 11, 1973, pp. 105-109.**[5]**S. A. ORSZAG & M. ISRAELI, "Numerical simulation of viscous incompressible flows,"*Annual Review of Fluid Mechanics*. Vol. VI, M. Van Dyke (Editor), Annual Reviews, 1974, pp. 281-318.**[6]**David M. Young,*Iterative solution of large linear systems*, Academic Press, New York-London, 1971. MR**0305568**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1975-0416049-2

Keywords:
Wave equation,
hyperbolic equations,
implicit difference scheme,
alternating direction method,
*L*-shaped domain,
higher order difference method

Article copyright:
© Copyright 1975
American Mathematical Society