Block Runge-Kutta methods for the numerical integration of initial value problems in ordinary differential equations. II. The stiff case

Author:
J. R. Cash

Journal:
Math. Comp. **40** (1983), 193-206

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679440-X

MathSciNet review:
679440

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Abstract: The approach described in the first part of this paper is extended to include diagonally implicit Runge-Kutta (DIRK) formulae. The algorithms developed are suitable for the numerical integration of stiff differential systems, and their efficiency is illustrated by means of some numerical examples.

**[1]**R. Alexander, "Diagonally implicit Runge-Kutta methods for stiff ordinary differential equations,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 1006-1021. MR**0458890 (56:17089)****[2]**J. Bond & J. R. Cash, "A block method for the numerical integration of stiff systems of ordinary differential equations,"*BIT*, v. 19, 1979, pp. 429-447. MR**559952 (81c:65032)****[3]**J. C. Butcher, "Coefficients for the study of Runge-Kutta integration processes,"*J. Austral. Math. Soc.*, v. 3, 1963, pp. 185-201. MR**0152129 (27:2109)****[4]**J. C. Butcher, K. Burrage & F. H. Chipman,*STRIDE*:*Stable Runge-Kutta Integrator for Differential Equations*, Report No. 20, Dept. of Mathematics, University of Auckland, New Zealand, 1979.**[5]**J. R. Cash, "Diagonally implicit Runge-Kutta formulae with error estimates,"*J. Inst. Math. Appl.*, v. 24, 1979, pp. 293-301.**[6]**J. R. Cash,*Stable Recursions, with Application to the Numerical Solution of Stiff Systems*, Academic Press, London and New York, 1979. MR**570113 (81j:65099)****[7]**M. Crouzieux,*Sur l'Approximation des équations Différentielles Opérationnelles Linéaires par des Méthodes de Runge-Kutta*, Ph.D. thesis, University of Paris, 1975.**[8]**C. W. Gear, "Runge-Kutta starters for multistep methods,"*ACM Trans. Math. Software*, v. 6, 1980, pp. 263-279. MR**585338 (81m:65119)****[9]**A. C. Hindmarsh,*GEAR*:*Ordinary Differential Equation System Solver*, Rep. UCID-30001, Rev. 3, Lawrence Livermore Laboratory, Livermore, Calif., 1974.**[10]**K. R. Jackson & R. Sacks-Davis, "An alternative implementation of variable step-size multistep formulas for stiff ODEs,"*ACM Trans. Math. Software*, v. 6, 1980, pp. 295-318. MR**585340 (81m:65120)****[11]**J. D. Lambert. Private communication, 1980.**[12]**B. Lindberg, "Characterization of optimal stepsize sequences for methods for stiff differential equations,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 859-887. MR**0519728 (58:24961)****[13]**W. E. Milne,*Numerical Solution of Differential Equations*, Wiley, New York, 1953. MR**0068321 (16:864c)****[14]**S. P. Nørsett,*Semi-Explicit Runge-Kutta Methods*, Mathematics and Computation, No. 6, University of Trondheim, 1974.**[15]**A. Prothero & A. Robinson, "On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations,"*Math. Comp.*, v. 28, 1974, pp. 145-162. MR**0331793 (48:10125)****[16]**L. F. Shampine & H. A. Watts, "*A*-stable implicit one-step methods,"*BIT*, v. 12, 1972, pp. 252-266. MR**0307483 (46:6603)****[17]**J. Williams & F. de Hoog, "A class of*A*-stable advanced multistep methods,"*Math. Comp.*, v. 28, 1974, pp. 163-177. MR**0356519 (50:8989)**

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DOI:
https://doi.org/10.1090/S0025-5718-1983-0679440-X

Article copyright:
© Copyright 1983
American Mathematical Society