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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
MathSciNet review: 0416049
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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,

$\displaystyle {U_{tt}} = a(x,y,t){U_{xx}} + b(x,y,t){U_{yy}},$

with fourth order accuracy in space and time. After an appropriate factorization, the resulting set of equations are tridiagonal and hence easily solved. The tridiagonal nature also indicates that the boundary conditions do not create special difficulties. Numerical experiments demonstrate the expected order of convergence and fulfill our expectations on the treatment of boundary conditions. An experimental computation also demonstrates that our results hold on L-shaped domains.

References [Enhancements On Off] (What's this?)

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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

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