New Step Size Control Algorithm for Semi-Implicit Composition ODE Solvers

Abstract

Composition is a powerful and simple approach for obtaining numerical integration methods of high accuracy order while preserving the geometric properties of a basic integrator. Adaptive step size control allows one to significantly increase the performance of numerical integration methods. However, there is a lack of efficient step size control algorithms for composition solvers due to some known difficulties in constructing a low-cost embedded local error estimator. In this paper, we propose a novel local error estimator based on a difference between the semi-implicit CD method and semi-explicit midpoint methods within a common composition scheme. We evaluate the performance of adaptive composition schemes with the proposed local error estimator, comparing it with the other state-of-the-art approaches. We show that composition ODE solvers with the proposed step size control algorithm possess higher numerical efficiency than known methods, by using a comprehensive set of nonlinear test problems.