A non-homogeneous method of third order for additive stiff systems of ordinary differential equations

TL;DR

This paper introduces a third-order additive method for solving stiff ODE systems that is L-stable, computationally efficient, and adaptable with automatic step size control, demonstrated to be reliable through numerical tests.

Contribution

A novel third-order additive method with explicit fourth stage, L-stability, and flexible Jacobian approximation, reducing computational costs compared to previous approaches.

Findings

Method is L-stable for the implicit part.

Numerical experiments confirm reliability and efficiency.

Automatic step size control enhances usability.

Abstract

In this paper we construct a third order method for solving additively split autonomous stiff systems of ordinary differential equations. The constructed additive method is L-stable with respect to the implicit part and allows to use an arbitrary approximation of the Jacobian matrix. In opposite to our previous paper, the fourth stage is explicit. So, the constructed method also has a good stability properties because of L-stability of the intermediate numerical formulas in the fourth stage, but has a lower computational costs per step. Automatic stepsize selection based on local error and stability control are performed. The estimations for error and stability control have been obtained without significant additional computational costs. Numerical experiments show reliability and efficiency of the implemented integration algorithm.