High Order Local Discontinuous Galerkin Methods for the Allen-Cahn Equation: Analysis and Simulation (Journal of Computational Mathematics, 2018)
In this paper, we present a local discontinuous Galerkin (LDG) method for the Allen-Cahn equation. We prove the energy stability, analyze the optimal convergence rate of $k + 1$ in $L^2$ norm and present the $(2k + 1)$-th order negative-norm estimate of the semi-discrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction (SDC) method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of $\mathcal{O}(h^{k+1})$ in $L^2$ norm and improve the LDG solution from $\mathcal{O}(h^{k+1})$ to $\mathcal{O}(h^{2k+1})$ with the accuracy enhancement post-processing technique.