Convergence theorem for difference approximations of hyperbolic quasilinear initialboundary value problems
Author:
Daniel Michelson
Journal:
Math. Comp. 49 (1987), 445459
MSC:
Primary 65N10; Secondary 65M10
MathSciNet review:
906181
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Dissipative difference approximations to multidimensional hyperbolic quasilinear initialboundary value problems are considered. The difference approximation is assumed to be consistent with the differential problem and its linearization should be stable in . A formal asymptotic expansion to the difference solution is constructed. This expansion includes boundary and initial layers. It is proved that the expansion indeed approximates the difference solution to the required order. As a result, the difference solution converges to the differential one as the mesh size h tends to 0.
 [1]
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
 [2]
HeinzOtto
Kreiss, Initial boundary value problems for hyperbolic
systems, Comm. Pure Appl. Math. 23 (1970),
277–298. MR 0437941
(55 #10862)
 [3]
Daniel
Michelson, Stability theory of difference
approximations for multidimensional initialboundary value
problems, Math. Comp. 40
(1983), no. 161, 1–45. MR 679433
(84d:65068), http://dx.doi.org/10.1090/S00255718198306794332
 [4]
Daniel
Michelson, Initialboundary value problems for incomplete singular
perturbations of hyperbolic systems, Largescale computations in fluid
mechanics, Part 2 (La Jolla, Calif., 1983), Lectures in Appl. Math.,
vol. 22, Amer. Math. Soc., Providence, RI, 1985,
pp. 127–132. MR 818784
(87b:76097)
 [5]
Jeffrey
B. Rauch and Frank
J. Massey III, Differentiability of solutions to
hyperbolic initialboundary value problems, Trans. Amer. Math. Soc. 189 (1974), 303–318. MR 0340832
(49 #5582), http://dx.doi.org/10.1090/S00029947197403408320
 [6]
Gilbert
Strang, Accurate partial difference methods. II. Nonlinear
problems, Numer. Math. 6 (1964), 37–46. MR 0166942
(29 #4215)
 [1]
 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)
 [2]
 H.O. Kreiss, "Initial boundary value problems for hyperbolic systems," Comm. Pure Appl. Math., v. 23, 1970, pp. 277298. MR 0437941 (55:10862)
 [3]
 D. Michelson, "Stability theory of difference approximations for multidimensional initialboundary value problems," Math. Comp., v. 40, 1983, pp. 145. MR 679433 (84d:65068)
 [4]
 D. Michelson, InitialBoundary Value Problems for Incomplete Singular Perturbations of Hyperbolic Systems, Lectures in Appl. Math., vol. 22, Amer. Math. Soc., Providence, R. I., 1985, pp. 127132. MR 818784 (87b:76097)
 [5]
 J. Rauch & F. Massey, "Differentiability of solutions to hyperbolic initialboundary value problems," Trans. Amer. Math. Soc., v. 189, 1974, pp. 303318. MR 0340832 (49:5582)
 [6]
 G. Strang, "Accurate partial difference methods II: Nonlinear problems," Numer. Math., v. 6, 1964, pp. 3746. MR 0166942 (29:4215)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65N10,
65M10
Retrieve articles in all journals
with MSC:
65N10,
65M10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198709061817
PII:
S 00255718(1987)09061817
Article copyright:
© Copyright 1987 American Mathematical Society
