Practical Construction of Two-Step Collocation Runge-Kutta Methods for Ordinary Differential Equations

Abstract

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.