Construction of high-order quadratically stable second-derivative general linear methods for the numerical integration of stiff ODEs

Theory of general linear methods (GLMs) for the numerical solution of autonomous system of ordinary differential equations of the form y′=f(y)y′=f(y) is extended to include the second derivative y″=g(y):=f′(y)f(y)y″=g(y):=f′(y)f(y). This extension of GLMs is called second derivative general linear methods (SGLMs). In this paper we will construct two-stage AA- and LL-stable SGLMs of order pp and stage order q=pq=p up to six with low error constants. We will show the efficiency of the proposed methods by implementing on some well-known stiff problems.