Stability theory of difference approximations for multidimensional initial-boundary value problems

Michelson, Daniel

doi:10.1090/S0025-5718-1983-0679433-2

Stability theory of difference approximations for multidimensional initial-boundary value problems
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Math. Comp. 40 (1983), 1-45 Request permission

Abstract:

A stability theory is developed for dissipative difference approximations to multidimensional initial-boundary value problems. The original differential problem should be strictly hyperbolic and the difference problem consistent with the differential one. An algebra of pseudo-difference operators is built and later used to prove the stability of the difference approximation with variable coefficients. In addition, stability of the Cauchy problem for weakly dissipative difference approximations with variable coefficients is proved.

References

Functional Anal. Appl.

L. S. Frank, Spaces of network functions, Mat. Sb. (N.S.) 86 (128) (1971), 187–233 (Russian). MR 0290583
L. S. Frank, Algèbre des opérateurs aux différences finies, Israel J. Math. 13 (1972), 24–55 (1973) (French, with English summary). MR 394283, DOI 10.1007/BF02760228
I. Gohberg and L. Rodman, On spectral analysis of non-monic matrix and operator polynomials. I. Reduction to monic polynomials, Israel J. Math. 30 (1978), no. 1-2, 133–151. MR 508259, DOI 10.1007/BF02760835
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 341888, DOI 10.1090/S0025-5718-1972-0341888-3
Bertil Gustafsson, The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comput. 29 (1975), 396–406. MR 0386296, DOI 10.1090/S0025-5718-1975-0386296-7
Heinz-Otto Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 277–298. MR 437941, DOI 10.1002/cpa.3160230304
Heinz-Otto Kreiss, Stability theory for difference approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714. MR 241010, DOI 10.1090/S0025-5718-1968-0241010-7
Heinz-Otto Kreiss, On difference approximations of the dissipative type for hyperbolic differential equations, Comm. Pure Appl. Math. 17 (1964), 335–353. MR 166937, DOI 10.1002/cpa.3160170306

Difference Approximations for Time Dependent Problems

Andrew Majda and Stanley Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math. 28 (1975), no. 5, 607–675. MR 410107, DOI 10.1002/cpa.3160280504
Andrew Majda, James McDonough, and Stanley Osher, The Fourier method for nonsmooth initial data, Math. Comp. 32 (1978), no. 144, 1041–1081. MR 501995, DOI 10.1090/S0025-5718-1978-0501995-4

Initial-Boundary Value Problems for Hyperbolic Equations and Their Difference Approximation With Uniformly Characteristic Boundary

Stanley Osher, Stability of difference approximations of dissipative type for mixed initial-boundary value problems. I, Math. Comp. 23 (1969), 335–340. MR 246530, DOI 10.1090/S0025-5718-1969-0246530-8
Beresford Parlett, Accuracy and dissipation in difference schemes, Comm. Pure Appl. Math. 19 (1966), 111–123. MR 196957, DOI 10.1002/cpa.3160190109
Jeffrey Rauch, ${\cal L}_{2}$ is a continuable initial condition for Kreiss’ mixed problems, Comm. Pure Appl. Math. 25 (1972), 265–285. MR 298232, DOI 10.1002/cpa.3160250305
Leonard Sarason, Hyperbolic and other symmetrizable systems in regions with corners and edges, Indiana Univ. Math. J. 26 (1977), no. 1, 1–39. MR 442495, DOI 10.1512/iumj.1977.26.26001