Error Analysis of an Upwind Weak Galerkin Finite Element Method for Time‐Dependent Hyperbolic Problems

In this article, we develop and analyze a fully discrete upwind weak Galerkin finite element method for solving time-dependent linear hyperbolic equations. The proposed numerical framework incorporates spatial discretization via the weak Galerkin method and employs both backward Euler and Crank-Nicolson schemes for temporal discretization. An upwind stabilization is introduced to handle discontinuities effectively. Stability and error estimates in both
and energy norms are established. The numerical results demonstrate that the method achieves expected accuracy and stability, validating the proposed scheme's robustness and effectiveness for practical applications involving time-dependent convection-dominated problems.