This paper analyzes the stability of the class of Time-Accurate and Highly-Stable Explicit Runge-Kutta (TASE-RK) methods, introduced in 2021 by Bassenne et al. (J. Comput. Phys.) for the numerical solution of stiff Initial Value Problems (IVPs). Such numerical methods are easy to implement and require the solution of a limited number of linear systems per step, whose coefficient matrices involve the exact Jacobian $J$ of the problem. To significantly reduce the computational cost of TASE-RK methods without altering their consistency properties, it is possible to replace $J$ with a matrix $A$ (not necessarily tied to $J$) in their formulation, for instance fixed for a certain number of consecutive steps or even constant. However, the stability properties of TASE-RK methods strongly depend on this choice, and so far have been studied assuming $A=J$. In this manuscript, we theoretically investigate the conditional and unconditional stability of TASE-RK methods by considering arbitrary $A$. To this end, we first split the Jacobian as $J=A+B$. Then, through the use of stability diagrams and their connections with the field of values, we analyze both the case in which $A$ and $B$ are simultaneously diagonalizable and not. Numerical experiments, conducted on Partial Differential Equations (PDEs) arising from applications, show the correctness and utility of the theoretical results derived in the paper, as well as the good stability and efficiency of TASE-RK methods when $A$ is suitably chosen.

翻译：本文分析了2021年Bassenne等人（J. Comput. Phys.）为数值求解刚性初值问题（IVPs）而引入的一类时间精确且高度稳定的显式龙格-库塔方法（TASE-RK）的稳定性。此类数值方法易于实现，每步仅需求解有限个线性系统，其系数矩阵涉及问题的精确雅可比矩阵$J$。为在不改变TASE-RK方法相容性性质的前提下显著降低计算成本，可在其公式中用矩阵$A$（不必与$J$相关）替代$J$，例如将$A$固定用于若干连续步骤甚至保持恒定。然而，TASE-RK方法的稳定性性质强烈依赖于该选择，且目前仅在假设$A=J$的条件下进行了研究。本文通过考虑任意矩阵$A$，从理论上探究了TASE-RK方法的条件稳定性和无条件稳定性。为此，我们首先将雅可比矩阵分解为$J=A+B$；随后，利用稳定性图及其与数值域的联系，分析了$A$与$B$可同时对角化及不可对角化两种情形。基于偏微分方程（PDEs）应用实例的数值实验验证了本文理论结果的正确性与实用性，并表明当$A$选择适当时，TASE-RK方法具有良好的稳定性和计算效率。