The unconditional instability of inflowdependent boundary conditions in difference approximations to hyperbolic systems
Author:
Eitan Tadmor
Journal:
Math. Comp. 41 (1983), 309319
MSC:
Primary 65M10
MathSciNet review:
717688
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Abstract: It is well known that for a mixed initialboundary hyberbolic system to be welldefined 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 zerothorder 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
(80a:65195), http://dx.doi.org/10.1090/S00255718197804920342
 [2]
Moshe
Goldberg and Eitan
Tadmor, Schemeindependent stability criteria
for difference approximations of hyperbolic initialboundary value
problems. II, Math. Comp.
36 (1981), no. 154, 603–626. MR 606519
(83f:65142), http://dx.doi.org/10.1090/S00255718198106065199
 [3]
Bertil
Gustafsson, The convergence rate for difference
approximations to mixed initial boundary value problems, Math. Comput. 29 (1975), 396–406. MR 0386296
(52 #7154), http://dx.doi.org/10.1090/S00255718197503862967
 [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, HeinzOtto
Kreiss, and Arne
Sundström, Stability theory of difference
approximations for mixed initial boundary value problems. II, Math. Comp. 26 (1972), 649–686. MR 0341888
(49 #6634), http://dx.doi.org/10.1090/S00255718197203418883
 [6]
A.
Harten, J.
M. Hyman, and P.
D. Lax, On finitedifference approximations and entropy conditions
for shocks, Comm. Pure Appl. Math. 29 (1976),
no. 3, 297–322. With an appendix by B. Keyfitz. MR 0413526
(54 #1640)
 [7]
Reuben
Hersh, Mixed problems in several variables, J. Math. Mech.
12 (1963), 317–334. MR 0147790
(26 #5304)
 [8]
HeinzOtto
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
(34 #7039)
 [9]
HeinzOtto
Kreiss, Stability theory for difference
approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714. MR 0241010
(39 #2355), http://dx.doi.org/10.1090/S00255718196802410107
 [10]
HeinzOtto
Kreiss and Einar
Lundqvist, On difference approximations with
wrong boundary values, Math. Comp. 22 (1968), 1–12. MR 0228193
(37 #3777), http://dx.doi.org/10.1090/S0025571819680228193X
 [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 lambdamatrices with
applications. II. Difference equations with constant coefficients,
Linear Algebra and Appl. 18 (1977), no. 3,
213–222. MR 0485917
(58 #5712)
 [13]
E. Tadmor, SchemeIndependent Stability Criteria for Difference Approximations to Hyperbolic InitialBoundary Value Systems, Ph.D. thesis, Dept. of Math. Sci., TelAviv Univ., December 1978.
 [14]
Lloyd
N. Trefethen, Group velocity in finite difference schemes,
SIAM Rev. 24 (1982), no. 2, 113–136. MR 652463
(83b:65141), http://dx.doi.org/10.1137/1024038
 [15]
Stanley
Osher, On systems of difference equations
with wrong boundary conditions, Math. Comp.
23 (1969),
567–572. MR 0247785
(40 #1046), http://dx.doi.org/10.1090/S00255718196902477856
 [1]
 A. Burns, "A necessary condition for the stability of a difference approximation to a hyperbolic system of partial differential equations," Math. Comp., v. 32, 1978, pp. 707724. MR 492034 (80a:65195)
 [2]
 M. Goldberg & E. Tadmor, "Schemeindependent stability criteria for difference approximations of hyperbolic initialboundary value problems. II," Math. Comp., v. 36, 1981, pp. 603626. MR 606519 (83f:65142)
 [3]
 B. Gustafsson, "The convergence rate for difference approximations to mixed initial boundary value problems," Math. Comp., v. 29, 1975, pp. 396406. MR 0386296 (52:7154)
 [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]
 B. Gustafsson, H.O. Kreiss & A. Sundström, "Stability theory of difference approximations for mixed initial boundary value problems. II," Math. Comp., v. 26, 1972, pp. 649686. MR 0341888 (49:6634)
 [6]
 A. Harten, J. M. Hyman & P. D. Lax, "On finite difference approximations and entropy conditions for shocks," Comm. Pure Appl. Math., v. 29, 1976, pp. 297322. MR 0413526 (54:1640)
 [7]
 R. Hersh, "Mixed problems in several variables," J. Math. Mech., v. 12, 1963, pp. 317334. MR 0147790 (26:5304)
 [8]
 H.O. Kreiss, "Difference approximations for hyperbolic differential equations," Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. of Maryland, 1965), Academic Press, New York, 1966, pp. 5158. MR 0207223 (34:7039)
 [9]
 H.O. Kreiss, "Stability theory for difference approximations of mixed initial boundary value problems. I," Math. Comp., v. 22, 1968, pp. 703714. MR 0241010 (39:2355)
 [10]
 H.O. Kreiss & E. Lundquist, "On difference approximations with wrong boundary values," Math. Comp., v. 22, 1968, pp. 112. MR 0228193 (37:3777)
 [11]
 H.O. Kreiss & J. Oliger, Methods for the Approximate Solution of Time Dependent Problems, GARP Publication Series No. 10, 1973.
 [12]
 P. Lancaster, "A fundamental theorm on lambdamatrices with applications. II. Finite difference equations," Linear Algebra Appl., v. 18, 1977, pp. 213222. MR 0485917 (58:5712)
 [13]
 E. Tadmor, SchemeIndependent Stability Criteria for Difference Approximations to Hyperbolic InitialBoundary Value Systems, Ph.D. thesis, Dept. of Math. Sci., TelAviv Univ., December 1978.
 [14]
 L. Trefethen, Group Velocity in Finite Difference Schemes, Report No. NA8112, Dept. of Comput. Sci., Stanford Univ., Stanford, 1981. MR 652463 (83b:65141)
 [15]
 S. Osher, "On systems of difference equations with wrong boundary conditions," Math. Comp., v. 23, 1969, pp. 567572. MR 0247785 (40:1046)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198307176886
PII:
S 00255718(1983)07176886
Article copyright:
© Copyright 1983
American Mathematical Society
