A Novel Second-Order and Unconditionally Energy Stable Numerical Scheme for Allen–Cahn Equation

Abstract

We propose a novel time-stepping scheme for solving the Allen–Cahn equation. We first rewrite the free energy into an equivalent form and then obtain a new Allen–Cahn equation by energy variational formula of $L^2$ -gradient flow. Using leapfrog formula, a new linear scheme is obtained, and we prove that the numerical scheme is unconditionally energy stable and uniquely solvable, and the discrete energy is in agreement with the original free energy. In addition, we also discuss the uniform boundedness and error estimate of numerical solution, the results show that the numerical solution is uniformly bounded in $H^2$ -norm, and error estimate shows that the time-stepping scheme can achieve second-order accuracy in time direction. At last, several numerical tests are illustrated to verify the theoretical results. The numerical strategy developed in this paper can be easily applied to other gradient flow models.