Time discretization in the backward solution of parabolic equations. I
HTML articles powered by AMS MathViewer
- by Lars Eldén PDF
- Math. Comp. 39 (1982), 53-68 Request permission
Abstract:
The problem of solving a parabolic partial differential equation backwards in time by a method related to the Tikhonov-Phillips regularization method is considered. Time discretizations based on Padé approximations of the exponential function are studied, and a priori estimates of the step length are given, which guarantee an almost optimal error bound. The computational efficiency of different discretizations is discussed. Some numerical examples are given. In Part II of this paper we study the backward beam method, and the same error estimates are obtained. A new scheme for time descretization based on Padé approximation is discussed.References
- S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math. 16 (1963), 121–239. MR 155203, DOI 10.1002/cpa.3160160204
- B. L. Buzbee and Alfred Carasso, On the numerical computation of parabolic problems for preceding times, Math. Comp. 27 (1973), 237–266. MR 368448, DOI 10.1090/S0025-5718-1973-0368448-3
- Lars Eldén, Regularization of the backward solution of parabolic problems, Inverse and improperly posed problems in differential equations (Proc. Conf., Math. Numer. Methods, Halle, 1979) Math. Res., vol. 1, Akademie-Verlag, Berlin, 1979, pp. 73–81. MR 536169
- Richard E. Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J. Math. Anal. 6 (1975), 283–294. MR 361447, DOI 10.1137/0506029
- Joel N. Franklin, Minimum principles for ill-posed problems, SIAM J. Math. Anal. 9 (1978), no. 4, 638–650. MR 498340, DOI 10.1137/0509044
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- P. M. Hummel and C. L. Seebeck Jr., A generalization of Taylor’s expansion, Amer. Math. Monthly 56 (1949), 243–247. MR 28907, DOI 10.2307/2304764
- Paolo Manselli and Keith Miller, Dimensionality reduction methods for efficient numerical solution, backward in time, of parabolic equations with variable coefficients, SIAM J. Math. Anal. 11 (1980), no. 1, 147–159. MR 556505, DOI 10.1137/0511013 V. A. Morozov, "On the restoration of functions with guaranteed accuracy," Numerical Analysis in Fortran, Moscow Univ. Press, Moscow, 1979, pp. 46-65. (Russian) H. Padé, "Sur la représentation approchée d’une fonctions par des fractions rationelles," Thesis, Ann. École Norm. (3), v. 9, 1892.
- L. E. Payne, Improperly posed problems in partial differential equations, Regional Conference Series in Applied Mathematics, No. 22, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0463736, DOI 10.1137/1.9781611970463
- David L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach. 9 (1962), 84–97. MR 134481, DOI 10.1145/321105.321114 V. N. Strakhov, "Solution of incorrectly-posed problems in Hilbert space," Differential Equations, v. 6, 1970, pp. 1136-1140. (Russian) A. N. Tikhonov, "Solution of incorrectly formulated problems and the regularization method," Dokl. Akad. Nauk SSSR, v. 151, 1963, pp. 501-504; English transl, in Soviet Math. Dokl., v. 4, 1963, pp. 1035-1038.
- Richard S. Varga, On higher order stable implicit methods for solving parabolic partial differential equations, J. Math. and Phys. 40 (1961), 220–231. MR 140191, DOI 10.1002/sapm1961401220
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Math. Comp. 39 (1982), 53-68
- MSC: Primary 65M30
- DOI: https://doi.org/10.1090/S0025-5718-1982-0658214-9
- MathSciNet review: 658214