README.md

2026CSZ

Codex/ChatGPT (July 2026)

Summary

The authors establish a unified error analysis for semidiscrete upwind discontinuous Galerkin schemes using total-degree $\mathbb P^k$ elements on uniform Cartesian meshes for two-dimensional linear variable-coefficient and nonlinear hyperbolic conservation laws. A specially designed projection and recursively constructed correction functions overcome the reduced degrees of freedom relative to tensor-product $\mathbb Q^k$ elements. For smooth solutions and a wind direction that does not change sign, the method attains optimal $L^2$ error $O(h^{k+1})$, while cell averages and downwind-edge averages superconverge at least as $O(h^{k+2})$. Numerical experiments for constant-coefficient, variable-coefficient, and nonlinear equations confirm the theory and show even higher cell-average rates in some cases. They also demonstrate that the $\mathbb P^k$ superconvergence disappears on perturbed nonuniform meshes.

Contributions

1. Resolved the optimal-convergence question for arbitrary-order $\mathbb P^k$ upwind DG methods on uniform Cartesian meshes.
2. Constructed a projection adapted to the limited degrees of freedom of total-degree polynomial spaces.
3. Developed correction functions yielding the optimal $L^2$ bound $O(h^{k+1})$.
4. Proved $O(h^{k+2})$ superconvergence for cell and downwind-edge averages under a fixed-sign wind condition.
5. Showed numerically that the superconvergence depends on uniform mesh structure and is generally lost on perturbed meshes.