Stability Analysis and Error Estimates of Semi-implicit Spectral Deferred Correction Coupled with Local Discontinuous Galerkin Method for Linear Convection–Diffusion Equations

Abstract

In this paper, we focus on the theoretical analysis of the second and third order semi-implicit spectral deferred correction (SDC) time discretization with local discontinuous Galerkin (LDG) spatial discretization for the one-dimensional linear convection–diffusion equations. We mainly study the stability and error estimates of the corresponding fully discrete scheme. Based on the Picard integral equation, the SDC method is driven iteratively by either the explicit Euler method or the implicit Euler method. It is easy to implement for arbitrary order of accuracy. For the semi-implicit SDC scheme, the iteration and the left-most endpoint involved in the integral for the implicit part increase the difficulty of the theoretical analysis. To be more precise, the test functions are more complex and the energy equations are more difficult to construct, compared with the Runge–Kutta type semi-implicit schemes. Applying the energy techniques, we obtain both the second and third order semi-implicit SDC time discretization with LDG spatial discretization are stable provided the time step $\tau \le \tau _{0}$, where the positive $\tau _{0}$ depends on the diffusion and convection coefficients and is independent of the mesh size h. We then obtain the optimal error estimates for the corresponding fully discrete scheme under the condition $\tau \le \tau _{0}$ with similar technique for stability analysis. Numerical examples are presented to illustrate our theoretical results.