This paper is concerned with the stability and error estimates of the local discontinuous Galerkin (LDG) method coupled with semi-implicit spectral deferred correction (SDC) time-marching up to third order accuracy for the Allen–Cahn equation. Since the SDC method is based on the first order convex splitting scheme, the implicit treatment of the nonlinear item results in a nonlinear system of equations at each step, which increases the difficulty of the theoretical analysis. For the LDG discretizations coupled with the second and third order SDC methods, we prove the unique solvability of the numerical solutions through the standard fixed point argument in finite dimensional spaces. At the same time, the iteration and integral involved in the semi-implicit SDC scheme also increase the difficulty of the theoretical analysis. Comparing to the Runge–Kutta type semi-implicit schemes which exclude the left-most endpoint, the SDC scheme in this paper includes the left-most endpoint as a quadrature node. This makes the test functions of the SDC scheme more complicated and the energy equations are more difficult to construct. We provide two different ideas to overcome the difficulty of the nonlinear terms. By choosing the test functions carefully, the energy stability and error estimates are obtained in the sense that the time step $\Delta t$ only requires a positive upper bound and is independent of the mesh size $h$. Numerical examples are presented to illustrate our theoretical results.

本文关注的是局部不连续Galerkin（LDG）方法与半隐式谱递延校正（SDC）时间步长达到Allen-Cahn方程的三阶精度的稳定性和误差估计。由于SDC方法基于一阶凸分裂方案，因此对非线性项的隐式处理会导致每个步骤的非线性方程组，这增加了理论分析的难度。对于结合二阶和三阶SDC方法的LDG离散化，我们通过有限维空间中的标准定点参数证明了数值解的唯一可解性。同时，半隐式SDC方案所涉及的迭代和积分也增加了理论分析的难度。与不包括最左端的Runge-Kutta型半隐式方案相比，本文的SDC方案将最左端作为正交节点。这使得SDC方案的测试功能更加复杂，能量方程式也更难以构建。我们提供了两种不同的想法来克服非线性项的困难。通过仔细选择测试功能，可以从时间步长的角度获得能量稳定性和误差估计 我们提供了两种不同的想法来克服非线性项的困难。通过仔细选择测试功能，可以从时间步长的角度获得能量稳定性和误差估计 我们提供了两种不同的想法来克服非线性项的困难。通过仔细选择测试功能，可以从时间步长的角度获得能量稳定性和误差估计$\Delta Ť$ 只需要一个正的上限，并且与网格大小无关 $H$。数值例子说明了我们的理论结果。