Modified TASE Runge–Kutta Methods for Integrating Stiff Differential Equations

We propose a new class of numerical methods, called MTRK for short, derived by an appropriate modification of the time-accurate and highly stable explicit (TASE) Runge-Kutta methods introduced by Bassenne, Fu, and Mani [J. Comput. Phys., 424 (2021), 109847] and then extended by Calvo, Montijano, and R\'andez [J. Comput. Phys., 436 (2021), 110316]. The MTRK methods are very efficient for dealing with the stiffness of differential problems without resorting to implicit methods, which incur high computational costs as they require the solution of nonlinear algebraic equations at each step. An in-depth analysis of the stability and consistency properties of MTRK methods via Butcher trees shows not only that they definitely improve existing TASE Runge-Kutta methods, but also that they can be advantageous compared to some well-known methods such as W-methods and Rosenbrock methods. This is confirmed by numerical experiments performed with nonlinear partial differential equations from applications.