The unconditional instability of inflow-dependent boundary conditions in difference approximations to hyperbolic systems

Author:
Eitan Tadmor

Journal:
Math. Comp. **41** (1983), 309-319

MSC:
Primary 65M10

MathSciNet review:
717688

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that for a mixed initial-boundary hyberbolic system to be well-defined it is necessary to impose *additional* boundary conditions only on the inflow eigenspace of the problem. We prove the discrete analogue of the above concerning difference approximations to such a system; that is, imposing numerical boundary conditions which are at least zeroth-order accurate with an *inflow* part of the interior equations leads to unconditional instability.

**[1]**Anne M. Burns,*A necessary condition for the stability of a difference approximation to a hyperbolic system of partial differential equations*, Math. Comp.**32**(1978), no. 143, 707–724. MR**492034**, 10.1090/S0025-5718-1978-0492034-2**[2]**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**, 10.1090/S0025-5718-1981-0606519-9**[3]**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**[4]**B. Gustafsson, H.-O. Kreiss & A. Sundström,*Stability Theory of Difference Approximations for Mixed Initial Boundary Value Problems*. II, Report No. 30, Dept. of Comput. Sci., Uppsala Univ., Uppsala, Sweden, 1971.**[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**, 10.1090/S0025-5718-1972-0341888-3**[6]**A. Harten, J. M. Hyman, and P. D. Lax,*On finite-difference approximations and entropy conditions for shocks*, Comm. Pure Appl. Math.**29**(1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR**0413526****[7]**Reuben Hersh,*Mixed problems in several variables*, J. Math. Mech.**12**(1963), 317–334. MR**0147790****[8]**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****[9]**Heinz-Otto Kreiss,*Stability theory for difference approximations of mixed initial boundary value problems. I*, Math. Comp.**22**(1968), 703–714. MR**0241010**, 10.1090/S0025-5718-1968-0241010-7**[10]**Heinz-Otto Kreiss and Einar Lundqvist,*On difference approximations with wrong boundary values*, Math. Comp.**22**(1968), 1–12. MR**0228193**, 10.1090/S0025-5718-1968-0228193-X**[11]**H.-O. Kreiss & J. Oliger,*Methods for the Approximate Solution of Time Dependent Problems*, GARP Publication Series No. 10, 1973.**[12]**Peter Lancaster,*A fundamental theorem on lambda-matrices with applications. II. Difference equations with constant coefficients*, Linear Algebra and Appl.**18**(1977), no. 3, 213–222. MR**0485917****[13]**E. Tadmor,*Scheme-Independent Stability Criteria for Difference Approximations to Hyperbolic Initial-Boundary Value Systems*, Ph.D. thesis, Dept. of Math. Sci., Tel-Aviv Univ., December 1978.**[14]**Lloyd N. Trefethen,*Group velocity in finite difference schemes*, SIAM Rev.**24**(1982), no. 2, 113–136. MR**652463**, 10.1137/1024038**[15]**Stanley Osher,*On systems of difference equations with wrong boundary conditions*, Math. Comp.**23**(1969), 567–572. MR**0247785**, 10.1090/S0025-5718-1969-0247785-6

Retrieve articles in *Mathematics of Computation*
with MSC:
65M10

Retrieve articles in all journals with MSC: 65M10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717688-6

Article copyright:
© Copyright 1983
American Mathematical Society