Convergence theorem for difference approximations of hyperbolic quasilinear initial-boundary value problems

Author:
Daniel Michelson

Journal:
Math. Comp. **49** (1987), 445-459

MSC:
Primary 65N10; Secondary 65M10

MathSciNet review:
906181

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Dissipative difference approximations to multi-dimensional hyperbolic quasi-linear initial-boundary 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, 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**[2]**Heinz-Otto Kreiss,*Initial boundary value problems for hyperbolic systems*, Comm. Pure Appl. Math.**23**(1970), 277–298. MR**0437941****[3]**Daniel Michelson,*Stability theory of difference approximations for multidimensional initial-boundary value problems*, Math. Comp.**40**(1983), no. 161, 1–45. MR**679433**, 10.1090/S0025-5718-1983-0679433-2**[4]**Daniel Michelson,*Initial-boundary value problems for incomplete singular perturbations of hyperbolic systems*, Large-scale 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****[5]**Jeffrey B. Rauch and Frank J. Massey III,*Differentiability of solutions to hyperbolic initial-boundary value problems*, Trans. Amer. Math. Soc.**189**(1974), 303–318. MR**0340832**, 10.1090/S0002-9947-1974-0340832-0**[6]**Gilbert Strang,*Accurate partial difference methods. II. Non-linear problems*, Numer. Math.**6**(1964), 37–46. MR**0166942**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N10,
65M10

Retrieve articles in all journals with MSC: 65N10, 65M10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0906181-7

Article copyright:
© Copyright 1987
American Mathematical Society