The validity of the modified equation for nonlinear shock waves. (English) Zbl 0581.76072

The modified (model, equivalent) equation is an important tool in designing and analyzing nonlinear difference schemes. In this note, the validity of this principle is rigorously established for nonlinear shock wave solutions and the upwind scheme in a particular case.


76L05 Shock waves and blast waves in fluid mechanics
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[1] Boris, J.; Book, D., J. Comput. Phys., 11, 38-69 (1973)
[2] Caflisch, R., Comm. Pure Appl. Math., 32, 531-548 (1979)
[3] Chin, R., J. Comput. Phys., 18, 233-247 (1975)
[4] Engquist, B.; Osher, S., Math. Comp., 34, 45-75 (1980)
[5] Feller, W., (An Introduction to Probability Theory and Its Applications, Vol. 1 (1950), Wiley: Wiley New York) · Zbl 0039.13201
[6] Harten, A., Comm. Pure Appl. Math., 30, 611-638 (1977)
[8] Harten, A.; Hyman, J.; Lax, P. D., Comm. Pure Appl. Math., 29, 297-322 (1976)
[9] Hedstrom, G., Math. Comp., 29, 969-977 (1975)
[10] Hirt, C., J. Comput. Phys., 2, 339-355 (1968)
[11] Illin, A. M.; Oleinik, O. A., Math. Sb., 51, 191-232 (1960)
[12] Jennings, G., Comm. Pure Appl. Math., 26, 25-37 (1973)
[13] Lax, P. D., Comm. Pure Appl. Math., 10, 537-566 (1957)
[14] Lax, P. D., Accuracy and resolution in the computation of solutions of linear and nonlinear equations, (Recent Advances in Numerical Analysis (1978), Academic Press: Academic Press New York/London), 107-117
[15] Lerat, A., Numerical shock structure and nonlinear corrections for difference schemes in conservation form, (Sixth International Conference on Numerical Methods in Fluid Dynamics. Sixth International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics No. 90 (1979), Springer-Verlag: Springer-Verlag New York/Berlin), 345-351
[16] Lerat, A.; Peyret, R., C. R. Acad. Sci. Paris Ser. A, 276, 759-762 (1973)
[17] Majda, A.; Osher, S., Numer. Math., 30, 429-452 (1978)
[18] Majda, A.; Ralston, J., Comm. Pure Appl. Math., 32, 445-482 (1979)
[20] Osher, S.; Ralston, J., Comm. Pure Appl. Math., 35, 737-749 (1982)
[21] Warming, R.; Hyett, B., J. Comput. Phys., 14, 159-179 (1974)
[22] Zwas, G.; Roseman, J., J. Comput. Phys., 12, 179-186 (1973)
[23] Lax, P. D., On difference schemes for solving initial value problems for conservation laws, (Proceedings, Rome Symposium on Questions in Numerical Analysis (June, 1958)) · Zbl 0057.32502
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