Error analysis of implicit-explicit weak Galerkin finite element method for time-dependent nonlinear convection-diffusion problem

Abstract

This paper focuses on the exploration of an implicit-explicit (IMEX) weak Galerkin finite element method (WG-FEM) applied to a one-dimensional nonlinear convection-diffusion equation. Based on a special weak form featuring two built-in parameters, we propose the fully implicit-explicit discrete WG finite element scheme. The diffusion term is treated implicitly, while the nonlinear convection term is treated explicitly. The WG-FEM utilizes locally piecewise polynomials of degree k to approximate the primal variable within the element interiors, along with piecewise polynomials of degree $k+1$ for the weak derivatives. Optimal error estimates in the $L^2$ norm for the fully discrete scheme are derived in the theoretical analysis. Furthermore, we conduct numerical experiments to illustrate the effectiveness and accuracy of the proposed scheme.