Models of difference schemes for by partial differential equations

Author:
G. W. Hedstrom

Journal:
Math. Comp. **29** (1975), 969-977

MSC:
Primary 65M15

DOI:
https://doi.org/10.1090/S0025-5718-1975-0388797-4

MathSciNet review:
0388797

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that difference schemes for hyperbolic equations display dispersion of waves. For a general dissipative difference scheme, we present a dispersive wave equation and show that the dispersions are essentially the same when the initial data is a step function.

**[1]**R. D. RICHTMEYER & K. W. MORTON,*Difference Methods for Initial-Value Problems*, 2nd ed., Wiley, New York, 1967. MR**36**#3515.**[2]**H. KREISS & J. OLIGER,*Methods for the Approximate Solution of Time Dependent Problems*, Global Atmospheric Research Programme, Publications Series, no. 10, Geneva, 1973.**[3]**C. W. HIRT, "Heuristic stability theory for finite-difference equations,"*J. Computational Phys.*, v. 2, 1968, pp. 339-355.**[4]**G. R. McGuire and J. Ll. Morris,*A class of second-order accurate methods for the solution of systems of conservation laws*, J. Computational Phys.**11**(1973), 531–549. MR**0331808****[5]**N. N. Janenko and Ju. I. Šokin,*The first differential approximation of difference schemes for hyperbolic systems of equations*, Sibirsk. Mat. Ž.**10**(1969), 1173–1187 (Russian). MR**0255080****[6]**Raymond C. Y. Chin,*Dispersion and Gibbs phenomenon associated with difference approximations to initial boundary-value problems for hyperbolic equations*, J. Computational Phys.**18**(1975), no. 3, 233–247. MR**0391530****[7]**Alain Lerat and Roger Peyret,*Sur l’origine des oscillations apparaissant dans les profils de choc calculés par des méthodes aux différences*, C. R. Acad. Sci. Paris Sér. A-B**276**(1973), A759–A762 (French). MR**0314276****[8]**Alain Lerat and Roger Peyret,*Sur le choix de schémas aux différences du second ordre fournissant des profils de choc sans oscillation*, C. R. Acad. Sci. Paris Sér. A-B**277**(1973), A363–A366 (French). MR**0337150****[9]**Philip Brenner and Vidar Thomée,*Estimates near discontinuities for some difference schemes*, Math. Scand.**28**(1971), 329–340 (1972). MR**0305613**, https://doi.org/10.7146/math.scand.a-11028**[10]**G. W. Hedstrom,*The rate of convergence of some difference schemes*, SIAM J. Numer. Anal.**5**(1968), 363–406. MR**0230489**, https://doi.org/10.1137/0705031**[11]**S. I. Serdjukova,*The oscillations that arise in numerical calculations of the discontinuous solutions of differential equations*, Ž. Vyčisl. Mat. i Mat. Fiz.**11**(1971), 411–424 (Russian). MR**0284018**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M15

Retrieve articles in all journals with MSC: 65M15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1975-0388797-4

Keywords:
Hyperbolic equations,
discontinuities,
models of difference schemes

Article copyright:
© Copyright 1975
American Mathematical Society