The Convergence of Semi-Implicit Numerical Methods

Rapid development of semi-implicit and semi-explicit integration techniques allowed to create relatively stable and efficient extrapolation and composition ODE solvers. However, there are several shortcomings in semi-implicit approach that should be taken into consideration while solving non-Hamiltonian systems. One of the most disturbing features of semi-implicit integration methods is their low convergence, which, in theory, can significantly affect the performance of the solver. In this paper we study the convergence of ODE solvers based on of semi-implicit integrators. The linear differential equations of different order are considered as a test systems. The dependence between method convergence and system order is revealed. The comparison with traditional ODE solvers is given. We experimentally show that the semi-implicit algorithms may exhibit a low convergence for a certain systems. We also propose a technique to reduce this effect - the introduction of correction coefficient and give an experimental evaluation of this approach.