High-Order and Energy-Stable Implicit-Explicit Relaxation Runge-Kutta Schemes for Gradient Flows

In this paper, we propose a class of high-order and energy-stable implicit-explicit relaxation Runge-Kutta (IMEX RRK) schemes for solving the phase-field gradient flow models. By incorporating the scalar auxiliary variable (SAV) method, the original equations are reformulated into equivalent forms, and the modified energy is introduced. Then, based on the reformulated equations, we propose a kind of IMEX RRK methods, which are rigorously proved to preserve the energy dissipation law and achieve high-order accuracy for both Allen-Cahn and Cahn-Hilliard equations. Numerical experiments are conducted to validate the theoretical results, including the accuracy of the approximate solution and the efficiency of the proposed scheme. Furthermore, the schemes are extended to multi-component gradient flows, with the vector-valued Allen-Cahn equations serving as a representative example.