On Runge-Kutta methods for parabolic problems with time-dependent coefficients

Author:
Ohannes A. Karakashian

Journal:
Math. Comp. **47** (1986), 77-101

MSC:
Primary 65N10; Secondary 65W05

DOI:
https://doi.org/10.1090/S0025-5718-1986-0842124-1

MathSciNet review:
842124

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Abstract: Galerkin fully discrete approximations for parabolic equations with time-dependent coefficients are analyzed. The schemes are based on implicit Runge-Kutta methods, and are coupled with preconditioned iterative methods to approximately solve the resulting systems of linear equations. It is shown that for certain classes of Runge-Kutta methods, the fully discrete equations exhibit parallel features that can be exploited to reduce the final execution time to that of a low-order method.

**[1]**O. Axelsson, "Solution of linear systems of equations: Iterative methods," in*Sparse Matrix Techniques*(V. A. Barker, ed.), Lecture Notes in Math., Vol. 572, Springer-Verlag, Berlin, Heidelberg and New York, 1976, pp. 1-51. MR**0448834 (56:7139)****[2]**G. A. Baker, J. H. Bramble & V. Thomée, "Single step Galerkin approximations for parabolic problems,"*Math. Comp.*, v. 31, 1977, pp. 818-847. MR**0448947 (56:7252)****[3]**L. A. Bales, "Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time-dependent coefficients,"*Math. Comp.*, v. 43, 1984, pp. 383-414. MR**758190 (86g:65179a)****[4]**L. A. Bales, O. A. Karakashian & S. M. Serbin, "On the -stability of rational approximations to the exponential function with only real poles." (Submitted for publication.)**[5]**J. H. Bramble & P. H. Sammon, "Efficient higher order single-step methods for parabolic problems: Part I,"*Math. Comp.*, v. 35, 1980, pp. 655-677. MR**572848 (81h:65110)****[6]**K. Burrage, "Efficiently implementable algebraically stable Runge-Kutta methods,"*SIAM J. Numer. Anal.*, v. 19, 1982, pp. 245-258. MR**650049 (83d:65235)****[7]**K. Burrage & J. C. Butcher, "Stability criteria for implicit Runge-Kutta methods,"*SIAM J. Numer. Anal.*, v. 16, 1979, pp. 46-57. MR**518683 (80b:65096)****[8]**J. C. Butcher, "Implicit Runge-Kutta processes,"*Math. Comp.*, v. 18, 1964, pp. 50-64. MR**0159424 (28:2641)****[9]**J. C. Butcher, "A stability property of implicit Runge-Kutta methods,"*BIT*, v. 15, 1975, pp. 358-361.**[10]**J. C. Butcher, "On the implementation of implicit Runge-Kutta methods,"*BIT*, v. 16, 1976, pp. 237-240. MR**0488746 (58:8263)****[11]**M. Crouzeix,*Sur l'Approximation des Équations Différentielles Opérationnelles Linéaires par des Méthodes de Runge-Kutta, Thèse*, Université de Paris VI, 1975.**[12]**M. Crouzeix, "Sur la*B*-stabilité des méthodes de Runge-Kutta,"*Numer. Math.*, v. 32, 1979, pp. 75-82. MR**525638 (80f:65081)****[13]**J. Douglas, Jr., T. Dupont & R. Ewing, "Incomplete iterations for time-stepping a Galerkin method for a quasilinear parabolic problem,"*SIAM J. Numer. Anal.*, v. 16, 1979, pp. 503-522. MR**530483 (80f:65117)****[14]**J. L. Lions & E. Magenes,*Nonhomogeneous Boundary Value Problems and Applications*, Vol. II, Springer-Verlag, New York, 1973. MR**0350179 (50:2672)****[15]**S. P. Nørsett, "One step methods of Hermite type for numerical integration of stiff systems,"*BIT*, v. 14, 1974, pp. 63-77. MR**0337014 (49:1787)****[16]**S. P. Nørsett & G. Wanner, "The real-pole sandwich for rational approximations and oscillation equations,"*BIT*, v. 19, 1979, pp. 79-94. MR**530118 (81d:65040)****[17]**P. H. Sammon,*Approximations for Parabolic Equations with Time-Dependent Coefficients*, Ph.D. Thesis, Cornell University, Ithaca, N.Y., 1978.**[18]**P. H. Sammon,*Convergence Estimates for Semidiscrete Parabolic Approximations*, Mathematics Research Center Technical Survey Report No. 2053, 1980.**[19]**A. Wolfbrandt, "A note on a recent result of rational approximations to the exponential function,"*BIT*, v. 17, 1977, pp. 367-368. MR**0464551 (57:4481)**

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0842124-1

Article copyright:
© Copyright 1986
American Mathematical Society