On Stability and Conservation Properties of (s)EPIRK Integrators in the Context of Discretized PDEs

Abstract

Exponential integrators are becoming increasingly popular for stiff problems of high dimension due to their attractive property of solving the linear part of the system exactly and hence being A-stable. In practice, however, exponential integrators are implemented using approximation techniques to matrix-vector products involving functions of the matrix exponential (the so-called $\varphi$-functions) to make them efficient and competitive to other state-of-the-art schemes. We will examine linear stability and provide a Courant–Friedrichs–Lewy (CFL) condition of special classes of exponential integrator schemes called EPIRK and sEPIRK and demonstrate their dependence on the parameters of the embedded approximation technique. Furthermore, a conservation property of the EPIRK schemes is proven.