An embedded fourth order method for solving structurally partitioned systems of ordinary differential equations

Abstract

An embedded one-step numerical method for structurally partitioned systems of ordinary differential equations (ODEs) is considered. Twoparametric families of methods of order four with automatic step-control are constructed for systems of ODEs of first and second order. The methods have fewer stages than classic Runge-Kutta methods.Mathematics Subject Classification: 65L05, 65L06

Embedded methods of order six for special systems of ordinary differential equations

“…Thus in [10] a method of order five with effective (using First Same as Last technique) six stages for y 0 and five stages for each y i and y j was constructed, while the classic method takes six stages for any unknown. For lower order methods same effect was also obtained [9].…”

“…Either group (2) or (3) can be missing (since the groups (3) and (4) are structurally identical we consider the case of group (4) missing to be the same). In the latter case even more efficient methods can be constructed [8,11]. And still greater advantage in necessary stages to order relation can be obtained when we have just a system with cross-dependency, which is to name the case when l = 1, n = 2 and the group (2) is absent.…”

Economic fourth order three-stage method for solving systems of second order differential equations with special structure

“…The method is denoted as RKS4(3)4F with the denotation RKSp(q)mF meaning Runge-Kutta type Structural method of order p with order q embedded estimator, with m stages and FSAL. It was modified so that the error estimation is made only for functions of the group (1). So only three stages are required for the group (2) and effectively the method has just three stages for all unknown functions.…”