$(m, k)$-schemes for stiff systems of ODEs and DAEs

This paper deals with the derivation of the optimal form of the Rosenbrock-type methods in terms of the number of non-zero parameters and computational costs per step. A technique of obtaining $(m, k)$-methods from the well-known Rosenbrock-type methods is justified. There are given formulas for the $(m, k)$-schemes parameters transformation for their two canonical representations and obtaining the form of a stability function. The authors have developed $L$-stable $(3, 2)$-method of order $3$ which requires two evaluations of a function: one evaluation of the Jacobian matrix and one $LU$-decomposition per step. Moreover, in this paper there is formulated an integration algorithm of the alternating step size based on $(3, 2)$-method. It provides the numerical solution for both explicit and implicit systems of ODEs. The numerical results confirming the efficiency of the new algorithm are given.