Galerkin/Runge-Kutta discretizations for parabolic equations with time-dependent coefficients

Author:
Stephen L. Keeling

Journal:
Math. Comp. **52** (1989), 561-586

MSC:
Primary 65N30; Secondary 65M60

MathSciNet review:
958873

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial-boundary value problems with time-dependent 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 time-independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low-order method.

**[1]**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****[2]**Laurence A. Bales,*Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time-dependent coefficients*, Math. Comp.**43**(1984), no. 168, 383–414. MR**758190**, 10.1090/S0025-5718-1984-0758190-6**[3]**James H. Bramble and Peter H. Sammon,*Efficient higher order single step methods for parabolic problems. I*, Math. Comp.**35**(1980), no. 151, 655–677. MR**572848**, 10.1090/S0025-5718-1980-0572848-X**[4]**J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin,*Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations*, SIAM J. Numer. Anal.**14**(1977), no. 2, 218–241. MR**0448926****[5]**J. C. Butcher,*Implicit Runge-Kutta processes*, Math. Comp.**18**(1964), 50–64. MR**0159424**, 10.1090/S0025-5718-1964-0159424-9**[6]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[7]**M. Crouzeix,*Sur l'Approximation des Équations Différentielles Opérationnelles Linéaires par des Méthodes de Runge-Kutta*, Thèse, Université de Paris VI, 1975.**[8]**K. Dekker and J. G. Verwer,*Stability of Runge-Kutta methods for stiff nonlinear differential equations*, CWI Monographs, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. MR**774402****[9]**Vassilios A. Dougalis and Ohannes A. Karakashian,*On some high-order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation*, Math. Comp.**45**(1985), no. 172, 329–345. MR**804927**, 10.1090/S0025-5718-1985-0804927-8**[10]**Jim Douglas Jr., Todd Dupont, and Richard E. Ewing,*Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem*, SIAM J. Numer. Anal.**16**(1979), no. 3, 503–522. MR**530483**, 10.1137/0716039**[11]**Louis A. Hageman and David M. Young,*Applied iterative methods*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. Computer Science and Applied Mathematics. MR**630192****[12]**E. Hairer and G. Wanner,*Algebraically stable and implementable Runge-Kutta methods of higher order*, SIAM J. Numer. Anal.**18**(1981), no. 6, 1098–1108. MR**639000**, 10.1137/0718074**[13]**Ohannes A. Karakashian,*On Runge-Kutta methods for parabolic problems with time-dependent coefficients*, Math. Comp.**47**(1986), no. 175, 77–101. MR**842124**, 10.1090/S0025-5718-1986-0842124-1**[14]**S. L. Keeling,*Galerkin/Runge-Kutta Discretizations for Parabolic Equations with Time Dependent Coefficients*, ICASE Report No. 87-61, NASA Langley Research Center, Hampton, VA, 1987.**[15]**S. L. Keeling,*Galerkin/Runge-Kutta Discretizations for Parabolic Partial Differential Equations*, Ph.D. Dissertation, University of Tennessee, 1986.**[16]**S. L. Keeling,*On Implicit Runge-Kutta Methods for Parallel Computations*, ICASE Report No. 87-58, NASA Langley Research Center, Hampton, VA, 1987.**[17]**P. H. Sammon,*Approximations for Parabolic Equations with Time-Dependent Coefficients*, Ph.D. Thesis, Cornell University, 1978.**[18]**Peter H. Sammon,*Convergence estimates for semidiscrete parabolic equation approximations*, SIAM J. Numer. Anal.**19**(1982), no. 1, 68–92. MR**646595**, 10.1137/0719002

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30,
65M60

Retrieve articles in all journals with MSC: 65N30, 65M60

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0958873-3

Keywords:
Implicit Runge-Kutta methods,
time-dependent coefficients,
error estimates,
order reduction

Article copyright:
© Copyright 1989
American Mathematical Society