B-series and Runge–Kutta methods

Abstract

The early Runge–Kutta methods were built around the aim of obtaining successively higher orders for a generic scalar problem. However, in modern computing there is no interest in numerical methods which are applicable only to scalar problems and, above order 4, an analysis based on B-series is more appropriate. Even for order 5 there exist scalar methods with reduced order when applied to non-scalar problems. Explicit methods to order 5 are derived. Order barriers are introduced through the simplest case (that order equal to the number of stages is impossible for explicit methods with greater than order 4). It is shown that this barrier can be circumvented through the use of effective order (conjugate order). Implicit methods, intended for the solution of stiff problems, are analysed and derived. Effective order is introduced and a new method of effective order 5 is derived.