Analysis of the Implicit-Explicit Ultra-Weak Discontinuous Galerkin Method for Convection-Diffusion Problems

Source abstract

In this paper, we first present the optimal error estimates of the semi-discrete ultra-weak discontinuous Galerkin method for solving one-dimensional linear convection-diffusion equations. Then, coupling with a kind of Runge-Kutta type implicit-explicit time discretization which treats the convection term explicitly and the diffusion term implicitly, we analyze the stability and error estimates of the corresponding fully discrete schemes. The fully discrete schemes are proved to be stable if the time-step $\tau ≤ \tau_0,$ where $\tau_0$ is a constant independent of the mesh-size $h.$ Furthermore, by the aid of a special projection and a careful estimate for the convection term, the optimal error estimate is also obtained for the third order fully discrete scheme. Numerical experiments are displayed to verify the theoretical results.