Strong Stability Preserving Second Derivative General Linear Methods with Runge–Kutta Stability

Abstract

In this paper we describe the construction of second derivative general linear method with Runge–Kutta stability property preserving the strong stability properties of spatial discretizations. Then we present such methods that are obtained by the solution of the constrained minimization problem with nonlinear inequality constraints, corresponding to the strong stability preserving property of these methods, and equality constraints, corresponding to the order, stage order and Runge–Kutta stability conditions. The derived methods are of order $$p=q$$ with $$r=2$$ and $$s=p$$ or $$s=p+1$$ , of order $$p=q=s=r-1$$ and of order $$p=q+1=s=r$$ , where q, s and r are the stage order, the number of internal and the number of external stages, respectively. Efficiency of the proposed methods together with verification of the order of convergence and capability of these methods in solving partial differential equations with smooth and discontinues initial data are shown by some numerical experiments.