Virtual element stabilization of convection-diffusion equation with shock capturing

Streamline upwind Petrov-Galerkin (SUPG) stabilization for the virtual element discretization of the convection-diffusion equation produces local oscillations in the crosswind direction. To overcome such a shortcoming, an additional shock-capturing term is added into the formulation. In this paper, we propose a nonlinear shock-capturing technique and prove the existence and stability of the discrete solution. The resulted nonlinear system of equations is solved using a simple iterative technique. Numerical experiments are conducted to show the efficiency of shock-capturing term in reducing the spurious oscillations along the crosswind direction.