Schemes of (m, k)-Type for Solving Differential-Algebraic and Stiff Systems

Abstract

A form of Rosenbrock-type methods optimal in terms of the number of non-zero parameters and computational costs per step is considered. A technique of obtaining $$(m, k)$$-methods from some well-known Rosenbrock-type methods is justified. Formulas for transforming the parameters of $$(m,k)$$-schemes and for obtaining a stability function are given for two canonical representations of the schemes. An $$L$$-stable $$(3, 2)$$-method of order 3 is proposed, which requires two evaluations of the function: one evaluation of the Jacobian matrix and one $$LU$$-decomposition per step. A variable step size integration algorithm based on the $$(3,2)$$-method is formulated. It provides a numerical solution for both explicit and implicit systems of ODEs. Numerical results are presented to show the efficiency of the new algorithm.