General Runge–Kutta TASE Methods for Reaction–Diffusion Problems

The class of Time Accurate and highly Stable Explicit Runge–Kutta (RK–TASE) methods has been introduced by Bassenne, Fu and Mani (J. Comput. Phys. 2021) and then extended by Calvo, Montijano and Rández (J. Comput. Phys. 2021), and Aceto, Conte and Pagano (Appl. Numer. Math. 2024), for the efficient solution of stiff initial value problems. With the aim of making RK–TASE methods suitable for the efficient solution of semi–discretized reaction–diffusion Partial Differential Equations (PDEs), in this work we exploit a general formulation of the schemes that allows to reduce both their computational cost and error constants, obtaining also good stability properties for such problems. In particular, a thorough study of the accuracy and stability properties leads to new RK–TASE methods up to order five that are more efficient than existing ones. Several experiments show that the new proposed RK–TASE methods are able to efficiently solve models of PDEs from applications that require integration over long time intervals. Furthermore, they show the better performance of the new RK–TASE compared to other numerical schemes from the scientific literature.