Ultra-weak discontinuous Galerkin method with IMEX-BDF time marching for two dimensional convection-diffusion problems

In this paper, we study the stability and error estimates of the fully discrete ultra-weak discontinuous Galerkin (UWDG) methods for solving two dimensional convection-diffusion problems, where the implicit-explicit backward difference formulas (IMEX-BDF) with order from one to five are considered in time discretization. By exploiting an extension of the multiplier technique applied in Wang et al. (2023) [41], and by utilizing the symmetry and coercivity properties of the UWDG discretization for the diffusion term, we establish a general framework of unconditional stability analysis for the fully discrete schemes. In addition, by exploiting the ultra-weak projection proposed in Chen and Xing (2024) [15], we obtain the optimal error estimates for the considered schemes. We also present some numerical results to verify the optimal accuracy of the considered schemes for both one and two dimensional convection-diffusion problems.