Stability of finitedifference models containing two boundaries or interfaces
Author:
Lloyd N. Trefethen
Journal:
Math. Comp. 45 (1985), 279300
MSC:
Primary 65M10
MathSciNet review:
804924
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Abstract: It is known that the stability of finitedifference models of hyperbolic initialboundary value problems is connected with the propagation and reflection of parasitic waves. Here the waves point of view is applied to models containing two boundaries or interfaces, where repeated reflection of trapped wave packets is a potential new source of instability. Our analysis accounts for various known instability phenomena in a unified way and leads to several new results, three of which are as follows. (1) Dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface. (2) Algebraic "GKS instabilities" can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present. (3) "GKSstability" and "Pstability" can be established in certain problems by showing that the numerical reflection coefficient matrices have norm less than one.
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 R. M. Beam, R. F. Warming & H. C. Yee, Stability Analysis for Numerical Boundary Conditions and Implicit Difference Approximations of Hyperbolic Equations, Proc. NASA Sympos. on Numerical Boundary Procedures, 1981, pp. 199207. MR 683521 (84g:65123)
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 M. Berger, Stability of Interfaces with Mesh Refinement, ICASE Report 8342.
 [4]
 D. Brown, "A note on the numerical solution of the wave equation with piecewise smooth coefficients," Math. Comp., v. 42, 1984, pp. 369391. MR 736442 (85h:65194)
 [5]
 G. Browning, H.O. Kreiss & J. Oliger, "Mesh refinement," Math. Comp., v. 27, 1973, pp. 2939. MR 0334542 (48:12861)
 [6]
 M. Ciment, "Stable matching of difference schemes," SIAM J. Numer. Anal., v. 9, 1972, pp. 695701. MR 0319383 (47:7927)
 [7]
 M. Giles, Eigenmode Analysis of Unsteady OneDimensional Euler Equations, NASA Contract Report 172217, ICASE 1983, submitted to AIAA J.
 [8]
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 H.O. Kreiss, "Initial boundary value problems for hyperbolic systems," Comm. Pure Appl. Math., v. 23, 1970, pp. 277298. MR 0437941 (55:10862)
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 H.O. Kreiss & J. Oliger, Methods for the Approximate Solution of Time Dependent Problems, Global Atmospheric Research Programme Publ. Series No. 10, Geneva, 1973.
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 J. Oliger, "Fourthorder difference methods for the initial boundaryvalue problem for hyperbolic equations," Math. Comp., v. 28, 1974, pp. 1525. MR 0359344 (50:11798)
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 J. Oliger, "Hybrid difference methods for the initial boundaryvalue problem for hyperbolic equations," Math. Comp., v. 30, 1976, pp. 724738. MR 0428727 (55:1747)
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 J. Oliger, "Constructing stable difference methods for hyperbolic equations," Numerical Methods for Partial Difference Equations (S. Parter, ed.), Academic Press, New York, 1979. MR 558221 (81k:65102)
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 S. Osher, "Systems of difference equations with general homogeneous boundary conditions," Trans. Amer. Math. Soc., v. 137, 1969, pp. 177201. MR 0237982 (38:6259)
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 J. C. South & M. M. Hafez, Stability Analysis of Intermediate Boundary Conditions in Approximate Factorization Schemes, Proc. 6th AIAA Computational Fluid Dynamics Conference, Danvers, MA, 1983.
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 J. A. Trapp & J. D. Ramshaw, "A simple heuristic method for analyzing the effects of boundary conditions on numerical stability," J. Comput. Phys., v. 20, 1976, pp. 238242.
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 L. N. Trefethen, Wave Propagation and Stability for Finite Difference Schemes, Ph. D. Thesis, Department of Computer Science, Stanford University, 1982.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198508049242
PII:
S 00255718(1985)08049242
Article copyright:
© Copyright 1985 American Mathematical Society
