Implicit-explicit second derivative general linear methods with strong stability preserving explicit part

Abstract

In this paper, we discuss the class of implicit–explicit (IMEX) methods for systems of ordinary differential equations where the explicit part has strong stability preserving (SSP) property, while the implicit part of the method has Runge–Kutta stability property together with A-, or L-stability. The explicit part of these methods is treated by the explicit second derivative general linear methods and the implicit part is treated by the implicit general linear methods. We construct methods of order p=s and stage order q=p and q=p−1 with r=2, of order p=q=s−1=r−1 and of order p=q+1=s−1=r−1 up to the order p=4 with large SSP coefficients with respect to the large region of absolute stability when the implicit and explicit parts interact with each other. In the case of methods with p=q=r−1=s−1 and p=q+1=s−1=r−1, the implicit part has inherent Runge–Kutta stability property, and is L-stable. Finally, we test the proposed SSP IMEX schemes on some one dimensional linear and nonlinear problems, and the results for the expected order of convergence are presented.