Stability analysis and error estimates of local discontinuous Galerkin methods with semi-implicit spectral deferred correction time-marching for the Allen–Cahn equation

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 only requires a positive upper bound and is independent of the mesh size

. Numerical examples are presented to illustrate our theoretical results.

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