Analysis of the Implicit-Explicit Ultra-Weak Discontinuous Galerkin Method for Convection-Diffusion Problems (Journal of Computational Mathematics, 2023)
In this paper, we first present the optimal error estimates of the semi-discrete ultra-weakdiscontinuous Galerkin method for solving one-dimensional linear convection-diffusion equations. Then, coupling with a kind of Runge-Kutta type implicit-explicit time discretizationwhich treats the convection term explicitly and the diffusion term implicitly, we analyzethe stability and error estimates of the corresponding fully discrete schemes. The fullydiscrete schemes are proved to be stable if the time-step $τ ≤ τ_0,$ where $τ_0$ is a constant independent of the mesh-size $h.$ Furthermore, by the aid of a special projection and a carefulestimate for the convection term, the optimal error estimate is also obtained for the thirdorder fully discrete scheme. Numerical experiments are displayed to verify the theoreticalresults.
Keywords: The ultra-weak discontinuous Galerkin method; Convection-diffusion; Implicit-explicit time discretization; Stability; Error estimate.