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.

翻译：Calvo等人在J.Sci. Comput. 2023中提出的单级TASE算子旨在降低Runge-Kutta-TASE（RKTASE）方法在通过$LU$分解求解相关线性系统时的计算成本。本文通过为Runge-Kutta方法的每个阶段考虑不同的TASE算子，提出对这些方法的改进以提高效率。我们证明所得RKTASE方法等价于$W$-方法（Steihaug和Wolfbrandt，Mathematics of Computation，1979），这使我们能够通过$W$-方法理论获得所提出的修正单级RKTASE方法（MSRKTASE）的阶条件。我们构建了二阶和三阶的新MSRKTASE方法，并通过线性和非线性刚性系统的数值实验验证了其有效性。结果表明，与先前的单级RKTASE格式相比，MSRKTASE格式显著提升了计算效率和精度。