Variable-stepsize doubly quasi-consistent singly diagonally implicit two-step peer pairs for solving stiff ordinary differential equations

Abstract

This paper aims at constructing the first variable-stepsize doubly quasi-consistent singly diagonally implicit two-step peer pairs equipped with local error control for accurate and efficient numerical integration of stiff ordinary differential equations (ODEs). The main novelty of the numerical integration tools under consideration is that these accommodate the property of double quasi-consistency to variable meshes. This property means that the principal terms of the local and global errors coincide. It improves greatly the accuracy of conventional local error control facility and even translates it into a global one when the influence of the remaining terms in the mentioned error expansions is negligible. Our study proves theoretical conditions, which ensure the double quasi-consistency in the family of variable-stepsize singly diagonally implicit two-step peer formulas, and presents a feasible procedure for constructing such numerical schemes in practice. In particular, two embedded doubly quasi-consistent pairs of convergence order 3 with local error control are built in this way. Our numerical examples confirm their high efficiency for treating stiff ODEs, including large-scale systems obtained by semidiscretization of partial differential equations (PDEs). Moreover, the numerical integration tools designed here are shown to outperform the well-optimized Matlab code ODE23s with local error control, which is also a stiff ODE solver of order 3. An exhaustive comparison to the other built-in Matlab code ODE15s with local error control, which is considered to be a benchmark means for solving stiff ODEs by many practitioners, is also fulfilled and commented.