Optimal Error Estimates of the Discontinuous Galerkin Method with Upwind-Biased Fluxes for 2D Linear Variable Coefficients Hyperbolic Equations

In this paper, we consider the discontinuous Galerkin method with upwind-biased numerical fluxes for two-dimensional linear hyperbolic equations with degenerate variable coefficients on Cartesian meshes. The \(L^2\)-stability is guaranteed by the numerical viscosity of the upwind-biased fluxes, and the adjustable numerical viscosity is useful in resolving waves and is beneficial for long time simulations. To derive optimal error estimates, a new projection is introduced and analyzed, which is the tensor product of the corresponding one-dimensional piecewise global projection for each variable. The analysis of uniqueness and optimal interpolation properties of the proposed projection is subtle, as the projection requires different collocations for the projection errors involving the volume integral, the boundary integral and the boundary points. By combining the optimal interpolation estimates and a sharp bound for the projection errors, optimal error estimates are obtained. Numerical experiments are shown to confirm the validity of the theoretical results.