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

Cash, J. R.

doi:10.1090/S0025-5718-1983-0679440-X

Block Runge-Kutta methods for the numerical integration of initial value problems in ordinary differential equations. II. The stiff case
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Math. Comp. 40 (1983), 193-206 Request permission

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.

References

Roger Alexander, Diagonally implicit Runge-Kutta methods for stiff o.d.e.’s, SIAM J. Numer. Anal. 14 (1977), no. 6, 1006–1021. MR 458890, DOI 10.1137/0714068
J. E. Bond and J. R. Cash, A block method for the numerical integration of stiff systems of ordinary differential equations, BIT 19 (1979), no. 4, 429–447. MR 559952, DOI 10.1007/BF01931259
J. C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc. 3 (1963), 185–201. MR 0152129, DOI 10.1017/S1446788700027932

STRIDE

Stable Runge-Kutta Integrator for Differential Equations

J. Inst. Math. Appl.

J. R. Cash, Stable recursions, Computational Mathematics and Applications, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York-Toronto, Ont., 1979. With applications to the numerical solution of stiff systems. MR 570113

Sur l’Approximation des équations Différentielles Opérationnelles Linéaires par des Méthodes de Runge-Kutta

C. W. Gear, Runge-Kutta starters for multistep methods, ACM Trans. Math. Software 6 (1980), no. 3, 263–279. MR 585338, DOI 10.1145/355900.355901

GEAR

Ordinary Differential Equation System Solver

K. R. Jackson and R. Sacks-Davis, An alternative implementation of variable step-size multistep formulas for stiff ODEs, ACM Trans. Math. Software 6 (1980), no. 3, 295–318. MR 585340, DOI 10.1145/355900.355903
Bengt Lindberg, Characterization of optimal stepsize sequences for methods for stiff differential equations, SIAM J. Numer. Anal. 14 (1977), no. 5, 859–887. MR 519728, DOI 10.1137/0714058
William Edmund Milne, Numerical solution of differential equations, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0068321

Semi-Explicit Runge-Kutta Methods

A. Prothero and A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comp. 28 (1974), 145–162. MR 331793, DOI 10.1090/S0025-5718-1974-0331793-2
H. A. Watts and L. F. Shampine, $A$-stable block implicit one-step methods, Nordisk Tidskr. Informationsbehandling (BIT) 12 (1972), 252–266. MR 307483, DOI 10.1007/bf01932819
Jack Williams and Frank de Hoog, A class of $A$-stable advanced multistep methods, Math. Comp. 28 (1974), 163–177. MR 356519, DOI 10.1090/S0025-5718-1974-0356519-8