Cell-vertex WENO schemes with shock-capturing quadrature for high-order finite element discretizations of hyperbolic problems

We propose a new kind of localized shock capturing for continuous (CG) and discontinuous Galerkin (DG) discretizations of hyperbolic conservation laws. The underlying framework of dissipation-based weighted essentially nonoscillatory (WENO) stabilization for high-order CG and DG approximations was introduced in our previous work. In this general framework, Hermite WENO (HWENO) reconstructions are used to calculate local smoothness sensors that determine the appropriate amount of artificial viscosity for each cell. In the original version, candidate polynomials for WENO averaging are constructed using the derivative data from von Neumann neighbors. We upgrade this standard ‘cell–cell’ reconstruction procedure by using WENO polynomials associated with mesh vertices as candidate polynomials for cell-based WENO averaging. The Hermite data of individual cells is sent to vertices of those cells, after which vertex-averaged HWENO data is sent back to cells containing the vertices. The new ‘cell-vertex’ averaging procedure includes the data of vertex neighbors without explicitly adding them to the reconstruction stencils. It mitigates mesh imprinting and can also be used in classical HWENO limiters for DG methods. The second main novelty of the proposed approach is a quadrature-driven distribution of artificial viscosity within high-order finite elements. Replacing the linear quadrature weights by their nonlinear WENO-type counterparts, we concentrate shock-capturing dissipation near discontinuities while minimizing it in smooth portions of troubled cells. This redistribution of WENO stabilization preserves the total dissipation rate within each cell and improves local shock resolution without relying on subcell decomposition techniques. Numerical experiments in one and two dimensions demonstrate substantial improvements in accuracy and robustness for high-order elements, providing a compelling alternative to standard cell–cell reconstructions and subcell shock-capturing schemes.