Economical Sixth Order Runge–Kutta Method for Systems of Ordinary Differential Equations

Abstract

Structural partitioning of systems of ordinary differential equations is made on base of right-hand side dependencies on the unknown variables. It is used to construct fully explicit Runge–Kutta methods with several computational schemes applied to different parts of the system. The constructed structural methods require fewer right-hand side evaluations (stages) per step for some parts of the system than classic explicit Runge–Kutta methods of the same order. The full structural form of the system is presented, which after permutation of variables can be applied to any system of ordinary differential equation. For such structure a multischeme method is formulated and conditions of the sixth order are written down. We present simplifying conditions and reduce the system to a solvable smaller system. A particular computational scheme, that requires seven stages for a group without special structure and only six stages for other equations, is presented. Its sixth order is confirmed by a numerical convergence test.