Algorithm of construction of effective explicit methods for structurally partitioned systems of ordinary differential equations

Abstract

Systems of ordinary differential equations partitioned on base of their right-hand side dependencies on the unknown functions are considered. Explicit multischeme Runge— Kutta methods for such systems are presented. These methods require fewer right-hand side computations (stages) than classic single-scheme Runge— Kutta methods to provide the same order of convergence. The full system of order conditions is presented. This system is reduced to several independent linear systems with help of the simplifying relations. The algorithm of computing the order conditions system solution with six free parameters is given. A particular choice of free parameters and the corresponding computational scheme are presented. The advantage of the presented methods is shown by the numerical comparison to the known classic six order method by J. C. Butcher.