中文摘要：

在这份报纸，我们在场为 Allen-Cahn 方程的一个本地不连续的 Galerkin (LDG ) 方法。我们证明精力稳定性，分析 k + 的最佳的集中率 1 在 L2 标准和现在(2k + 1 )-th 顺序为有光滑的答案的 Allen-Cahn 方程的 semidiscrete LDG 方法的否定标准的估计。为了放松严重时间，走行进方法的明确的时间的限制，我们构造第一基于分离 Allen-Cahn 精力的凸的切开原则订半含蓄的计划并且证明相应无条件的精力稳定性。为了高完成，命令时间的精确性，我们采用半含蓄光谱推迟的修正(SDC ) 方法。与无条件地稳定的凸的切开计划结合， SDC 方法能是在我们的数字测试精确、稳定的高顺序。提高建议方法的效率， multigrid 解答者被改编解决结果非线性的代数学的系统。数字研究被介绍证实我们能在 L2 标准完成 O (hk+1 ) 的最佳的精确性并且与精确性改进 processing 以后技术从 O (hk+1 ) 改进 LDG 答案到 O (h2k+1 ) 。 '

英文摘要：

In this paper, we present a local discontinuous Galerkin （LDG） method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k ＋ 1 in L2 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 （O（hk＋1） in L2 norm and improve the LDG solution from （O（hk＋1） to （O（h2k＋1） with the accuracy enhancement post-processing technique.