A new class of split exponential propagation iterative methods of Runge–Kutta type (sEPIRK) for semilinear systems of ODEs

In this paper the framework of the exponential propagation iterative methods of Runge–Kutta type (EPIRK) is extended to construct split EPIRK (sEPIRK) integrators for semilinear systems of ODEs. The structure of the sEPIRK methods possesses the flexibility and generality that allows construction of very efficient schemes. We demonstrate how bicolored trees-based B-series can be used to derive the order conditions for the new integrators. An algorithm is developed to solve the order conditions up to order three and several new schemes with desirable properties are proposed. The numerical results illustrate the advantages offered by the new class of integrators. The experiments also address the comparative performance of split vs. non-split EPIRK methods and the question of improving efficiency by optimizing coefficients of the sEPIRK schemes. It is shown that specific schemes can be custom-built to improve computational efficiency for a particular problem.