Error analysis of finite difference schemes applied to hyperbolic initial-boundary value problems

Author:
Gunilla Sköllermo

Journal:
Math. Comp. **33** (1979), 11-35

MSC:
Primary 65N15

MathSciNet review:
514808

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Abstract: A technique for the complete error analysis of finite difference schemes for hyperbolic initial boundary value problems is developed. The error analysis is split into subproblems so that different boundary approximations or different initial approximations easily can be compared for a given interior scheme. The steps of the theoretical analysis are demonstrated on the leapfrog scheme for a simple model equation. The technique is applied to several choices of initial approximations and boundary conditions for leapfrog with second and fourth order accuracy in space. A comparison with two implicit schemes is also made. The theoretical error estimates are shown to agree very well with computational results.

**[1]**T. ELVIUS & A. SUNDSTRÖM, "Computationally efficient schemes and boundary conditions for a fine-mesh barotropic model based on the shallow-water equations,"*Tellus*, v. 25, 1973, pp. 132-156.**[2]**Bertil Gustafsson,*The convergence rate for difference approximations to mixed initial boundary value problems*, Math. Comput.**29**(1975), 396–406. MR**0386296**, 10.1090/S0025-5718-1975-0386296-7**[3]**Bertil Gustafsson, Heinz-Otto Kreiss, and Arne Sundström,*Stability theory of difference approximations for mixed initial boundary value problems. II*, Math. Comp.**26**(1972), 649–686. MR**0341888**, 10.1090/S0025-5718-1972-0341888-3**[4]**Heinz-Otto Kreiss and Joseph Oliger,*Comparison of accurate methods for the integration of hyperbolic equations*, Tellus**24**(1972), 199–215 (English, with Russian summary). MR**0319382****[5]**Joseph Oliger,*Fourth order difference methods for the initial boundary-value problem for hyperbolic equations*, Math. Comp.**28**(1974), 15–25. MR**0359344**, 10.1090/S0025-5718-1974-0359344-7**[6]**G. SKÖLLERMO,*How the Boundary Conditions Affect the Stability and Accuracy of Some Implicit Methods for Hyperbolic Equations*, Report No. 62, Dept. of Comput. Sci., Uppsala University, 1975.**[7]**Blair Swartz and Burton Wendroff,*The relative efficiency of finite difference and finite element methods. I. Hyperbolic problems and splines*, SIAM J. Numer. Anal.**11**(1974), 979–993. MR**0362952**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1979-0514808-2

Keywords:
Error analysis,
mixed initial boundary value problem

Article copyright:
© Copyright 1979
American Mathematical Society