Galerkin/RungeKutta discretizations for parabolic equations with timedependent coefficients
Author:
Stephen L. Keeling
Journal:
Math. Comp. 52 (1989), 561586
MSC:
Primary 65N30; Secondary 65M60
MathSciNet review:
958873
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Abstract: A new class of fully discrete Galerkin/RungeKutta methods is constructed and analyzed for linear parabolic initialboundary value problems with timedependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order of convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved and computational results are presented. Additionally, since the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximately. Nevertheless, the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with timeindependent coefficients. Furthermore, it is noted that special RungeKutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a loworder method.
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 R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
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 L. A. Bales, "Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with timedependent coefficients," Math. Comp., v. 43, 1984, pp. 383414. MR 758190 (86g:65179a)
 [3]
 J. H. Bramble & P. H. Sammon, "Efficient higher order single step methods for parabolic problems: Part I," Math. Comp., v. 35, 1980, pp. 655677. MR 572848 (81h:65110)
 [4]
 J. H. Bramble, A. H. Schatz, V. Thomée & L. B. Wahlbin, "Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations," SIAM J. Numer. Anal., v. 14, 1977, pp. 218241. MR 0448926 (56:7231)
 [5]
 J. C. Butcher, "Implicit RungeKutta processes," Math. Comp., v. 18, 1964, pp. 5064. MR 0159424 (28:2641)
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 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
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 M. Crouzeix, Sur l'Approximation des Équations Différentielles Opérationnelles Linéaires par des Méthodes de RungeKutta, Thèse, Université de Paris VI, 1975.
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 K. Dekker & J. G. Verwer, Stability of RungeKutta Methods for Stiff Nonlinear Differential Equations, NorthHolland, Amsterdam, 1984. MR 774402 (86g:65003)
 [9]
 V. A. Dougalis & O. A. Karakashian, "On some highorder accurate fully discrete Galerkin methods for the KortewegDe Vries equation," Math. Comp., v. 45, 1985, pp. 329345. MR 804927 (86m:65118)
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 [16]
 S. L. Keeling, On Implicit RungeKutta Methods for Parallel Computations, ICASE Report No. 8758, NASA Langley Research Center, Hampton, VA, 1987.
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 P. H. Sammon, Approximations for Parabolic Equations with TimeDependent Coefficients, Ph.D. Thesis, Cornell University, 1978.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909588733
PII:
S 00255718(1989)09588733
Keywords:
Implicit RungeKutta methods,
timedependent coefficients,
error estimates,
order reduction
Article copyright:
© Copyright 1989
American Mathematical Society
