Explicit Runge-Kutta scheme with high efficiency and energy dissipation for the Allen-Cahn equation

Abstract

This paper presents a series of explicit Runge-Kutta solution to the Allen-Cahn equation, using the same energy and the square of the newly developed method were analyzed. Based on the explicit Runge-Kutta, we find that these schemes not only have high accuracy, high-order convergence and efficiency, but also have energy-consuming characteristics. We first convert the nonlinear Allen-Cahn equation into an equivalent equation that satisfies the law of quadratic energy dissipation. Then, we discretize the reconstruction by using the Runge-Kutta scheme in the time direction. Theoretical analysis and numerical simulation results verify the algorithm’s dissipation characteristics, convergence order and long-term simulation stability.