A generalization of Runge-Kutta methods

M. Sofroniou, P.C. Moan, G.R.W. Quispel, G. Spaletta

Wolfram Research, 100 Trade Center Drive, Champaign, Illinois, U. S. A

marks@wolfram.com

Contributed talk

In this talk we will give an overview of a novel extension of Runge-Kutta methods for numerically solving systems of differential equations. The new schemes, which we have called Elementary Differential Runge-Kutta methods, evaluate elementary differentials in the internal stages. In particular, EDRK methods include the following schemes as a subset:

• Runge-Kutta methods
• Taylor series methods
• Multiderivative methods (see [6]).

We will outline how order conditions have been constructed for the new schemes using B-Series and their composition [1,3,2]. Some properties of EDRK methods are given including a categorization of symmetric methods. Symplectic numerical methods are advantageous for solving Hamiltonian systems of differential equations (see for example [7,4]). It is known that multiderivative methods cannot be symplectic [5]. In contrast, we will demonstrate that symplectic EDRK schemes exist. Following the approach in [8] we illustrate how simplified order conditions for symplectic methods have been derived. Finally we conclude with some numerical experiments that confirm the theory that has been developed.