Two classes of internally -stable generalized Runge-Kutta processes which remain consistent with an inaccurate Jacobian

Authors:
J. D. Day and D. N. P. Murthy

Journal:
Math. Comp. **39** (1982), 491-509

MSC:
Primary 65L20

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669642-X

MathSciNet review:
669642

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Abstract: Generalized Runge-Kutta Processes for stiff systems of ordinary differential equations usually require an accurate evaluation of a Jacobian at every step. However, it is possible to derive processes which are Internally *S*-stable when an accurate Jacobian is used but still remain consistent and highly stable if an approximate Jacobian is used. It is shown that these processes require at least as many function evaluations as an explicit Runge-Kutta process of the same order, and second and third order processes are developed. A second class of Generalized Runge-Kutta is introduced which requires that the Jacobian be evaluated accurately less than once every step. A third order process of this class is developed, and all three methods contain an error estimator similar to those of Fehlberg or England.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669642-X

Keywords:
Generalized Runge-Kutta procedure,
semi-implicit Runge-Kutta procedure,
approximate Jacobian,
stiff differential equations,
*L*-stability,
*A*-stability,
*S*-stability,
Internal *S*-stability

Article copyright:
© Copyright 1982
American Mathematical Society