Implicit–explicit second derivative diagonally implicit multistage integration methods

Abstract

We introduce a class of implicit–explicit (IMEX) schemes for the numerical solution of initial value problems of differential equations with both non-stiff and stiff components in which non-stiff and stiff solvers are, respectively, based on the explicit general linear methods (GLMs) and implicit second derivative GLMs (SGLMs). The order conditions of the proposed IMEX schemes are obtained. Linear stability properties of the methods are analyzed and then methods up to order four with a large area of absolute stability region of the pair are constructed assuming that the implicit part of the methods is L-stable. Due to the high stage orders of the constructed methods, they are not marred by order reduction. This is verified by the numerical experiments which demonstrate the efficiency of the proposed methods, too.