Modified Singly-Runge-Kutta-TASE methods for the numerical solution of stiff differential equations

TL;DR

This paper introduces modified Runge-Kutta-TASE methods with different TASE operators per stage, improving efficiency and accuracy in solving stiff differential equations, validated through numerical experiments.

Contribution

It proposes a new class of MSRKTASE methods, establishing their equivalence to W-methods and deriving their order conditions for better performance.

Findings

MSRKTASE methods of order two and three successfully implemented

Numerical experiments show improved efficiency over previous schemes

Enhanced accuracy in solving linear and nonlinear stiff systems

Abstract

Singly-TASE operators for the numerical solution of stiff differential equations were proposed by Calvo et al. in J.Sci. Comput. 2023 to reduce the computational cost of Runge-Kutta-TASE (RKTASE) methods when the involved linear systems are solved by some $LU$ factorization. In this paper we propose a modification of these methods to improve the efficiency by considering different TASE operators for each stage of the Runge-Kutta. We prove that the resulting RKTASE methods are equivalent to $W$-methods (Steihaug and Wolfbrandt, Mathematics of Computation,1979) and this allows us to obtain the order conditions of the proposed Modified Singly-RKTASE methods (MSRKTASE) through the theory developed for the $W$-methods. We construct new MSRKTASE methods of order two and three and demonstrate their effectiveness through numerical experiments on both linear and nonlinear stiff systems. The results show that the MSRKTASE schemes significantly enhance efficiency and accuracy compared to previous Singly-RKTASE schemes.