A general class of second-order [formula omitted]-stable explicit numerical methods for stiff problems

Abstract

In this paper, we propose a simple approach to the construction of a general class of L-stable explicit second-order one-step methods for solving stiff problems. These methods are nonlinear and derive from a novel approximation for the right-hand side functions of differential equations inspired by the nonstandard finite difference methodology introduced by Mickens. Through rigorous mathematical analysis, it is proved that the proposed numerical methods are not only explicit and L-stable, but also convergent of order two. Therefore, they are suitable and efficient to solve stiff problems.The proposed numerical methods generalize and improve a nonstandard explicit integration scheme for initial value problems formulated by Ramos (2007). Moreover, the present approach can be extended to construct A-stable and L-stable high-order explicit one-step methods for differential equations.Finally, the theoretical findings and advantages of the developed numerical methods are supported and illustrated by a series of numerical experiments in which stiff problems are considered.