Transformed implicit-explicit second derivative diagonally implicit multistage integration methods with strong stability preserving explicit part

Abstract

In this paper, we discuss the construction of a class of implicit-explicit (IMEX) methods for systems of ordinary differential equations which their right hand side can be split into two parts; nonstiff or mildly stiff part and stiff part. The proposed methods treat the non-stiff part by an explicit second derivative diagonally implicit multistage integration method (SDIMSIM) and the stiff part by an implicit diagonally implicit multistage integration method (DIMSIM). The explicit part of these methods has strong stability preserving (SSP) property and the implicit part is A- and L-stable. We will construct methods with p=q=r=s and p=q+1=r=s up to order four with large SSP coefficients with respect to the large region of absolute stability, assuming that the implicit part of the method has Runge–Kutta stability (RKS) property together with A- and L-stability. These methods are tested on the linear advection-diffusion, advection-reaction and nonlinear shallow water equations, and the numerical results are presented conforming the efficiency and order of constructed methods.