Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems. II

Authors:
Moshe Goldberg and Eitan Tadmor

Journal:
Math. Comp. **48** (1987), 503-520

MSC:
Primary 65M10

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878687-0

MathSciNet review:
878687

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Abstract: The purpose of this paper is to extend the results of [4] in order to achieve more versatile, convenient stability criteria for a wide class of finite-difference approximations to initial-boundary value problems associated with the hyperbolic system in the quarter plane , . With these criteria, stability is easily established for a large number of examples, where many of the cases studied in the recent literature are included and generalized.

**[1]**Moshe Goldberg,*On a boundary extrapolation theorem by Kreiss*, Math. Comp.**31**(1977), no. 138, 469–477. MR**0443363**, https://doi.org/10.1090/S0025-5718-1977-0443363-9**[2]**Moshe Goldberg and Eitan Tadmor,*Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. I*, Math. Comp.**32**(1978), no. 144, 1097–1107. MR**501998**, https://doi.org/10.1090/S0025-5718-1978-0501998-X**[3]**Moshe Goldberg and Eitan Tadmor,*Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II*, Math. Comp.**36**(1981), no. 154, 603–626. MR**606519**, https://doi.org/10.1090/S0025-5718-1981-0606519-9**[4]**Moshe Goldberg and Eitan Tadmor,*Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems*, Math. Comp.**44**(1985), no. 170, 361–377. MR**777269**, https://doi.org/10.1090/S0025-5718-1985-0777269-7**[5]**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**, https://doi.org/10.1090/S0025-5718-1972-0341888-3**[6]**Bertil Gustafsson and Joseph Oliger,*Stable boundary approximations for implicit time discretizations for gas dynamics*, SIAM J. Sci. Statist. Comput.**3**(1982), no. 4, 408–421. MR**677095**, https://doi.org/10.1137/0903026**[7]**Heinz-Otto Kreiss,*Difference approximations for hyperbolic differential equations*, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 51–58. MR**0207223****[8]**H.-O. Kreiss & J. Oliger,*Methods for the Approximate Solution of Time Dependent Problems*, GARP Publication Series No. 10, Geneva, 1973.**[9]**Joseph Oliger,*Fourth order difference methods for the initial boundary-value problem for hyperbolic equations*, Math. Comp.**28**(1974), 15–25. MR**0359344**, https://doi.org/10.1090/S0025-5718-1974-0359344-7**[10]**Stanley Osher,*Systems of difference equations with general homogeneous boundary conditions*, Trans. Amer. Math. Soc.**137**(1969), 177–201. MR**0237982**, https://doi.org/10.1090/S0002-9947-1969-0237982-4**[11]**Stanley Osher,*Stability of parabolic difference approximations to certain mixed initial boundary value problems*, Math. Comp.**26**(1972), 13–39. MR**0298990**, https://doi.org/10.1090/S0025-5718-1972-0298990-4**[12]**G. Sköllermo,*How the Boundary Conditions Affect the Stability and Accuracy of Some Methods for Hyperbolic Equations*, Report No. 62, Dept. of Computer Science, Uppsala University, Uppsala, Sweden, 1975.**[13]**G. Sköllermo,*Error Analysis for the Mixed Initial Boundary Value Problem for Hyperbolic Equations*, Report No. 63, Dept. of Computer Science, Uppsala University, Uppsala, Sweden, 1975.**[14]**E. Tadmor,*Scheme-Independent Stability Criteria for Difference Approximations to Hyperbolic initial-Boundary Value Systems*, Ph.D. thesis, Dept. of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel, 1978.**[15]**L. N. Trefethen,*Wave Propagation and Stability for Finite Difference Schemes*, Ph.D. thesis, Report No. STAN-CS-82-905, Computer Science Department, Stanford University, Stanford, California, 1982.

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DOI:
https://doi.org/10.1090/S0025-5718-1987-0878687-0

Article copyright:
© Copyright 1987
American Mathematical Society