Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems (Q2820344)

\textit{H. Wang} et al. [SIAM J. Numer. Anal. 53, No. 1, 206--227 (2015; Zbl 1327.65179); ``Stability and error estimates of the local discontinuous Galerkin method with implicit-explicit time-marching for nonlinear convection-diffusion problems'', Appl. Math. Comput. 272, 237--258 (2015)], had shown that the three specific Runge-Kutta type implicit-explicit (IMEX) time discretizations schemes, coupled with local discontinuous Galerkin (LDG) spatial discretization for solving one-dimensional linear and nonlinear convection-diffusion problems, are unconditional stable in the sense that the time step is only required to be upper bounded by a constant which is independent of the mesh size. In this paper, the authors show that the same stability holds for the IMEX LDG schemes considered in [loc. cit.] for solving multi-dimensional nonlinear convection-diffusion problems, on both rectangular meshes and triangular meshes. By using the so-called elliptic projection and the adjoint argument, they also obtain optimal error estimates for the corresponding fully discrete IMEX LDG schemes under the same condition as the stability analysis. Numerical examples are also given to verify the main results presented here.