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

TL;DR

This work tackles the non-oscillatory discretization of hyperbolic conservation laws by introducing a cell-vertex HWENO reconstruction that integrates vertex-neighborhood information without enlarging stencils, paired with a quadrature-driven redistribution of artificial viscosity to localize dissipation near discontinuities. The stabilization blends high- and low-order FE terms through a smoothness sensor, and the vertex-based HWENO candidates are blended with the cell candidate to form robust, high-order approximations for CG and DG methods. Numerical experiments in 1D and 2D, including Euler equations, demonstrate improved accuracy and robustness, with entropy-stable-like behavior observed numerically and potential for invariant-domain preservation via convex limiting. The proposed approach reduces mesh imprinting, avoids subcell decompositions, and provides a scalable, accurate framework for high-order hyperbolic problems across CG and DG discretizations.

Abstract

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.

Figures (9)

• Figure 1: Candidate polynomial stencils for triangular elements. The left two figures show cell-cell stencils. The first one illustrates the cell-based stencil for a single candidate polynomial, while the second one displays the full stencil of a cell-cell reconstruction. The right two figures present the corresponding vertex-based and cell-vertex stencils. In both approaches, four candidate polynomials are used, including the finite element approximation $u_h^e$ in the target cell $K_e$.
• Figure 2: Normalized scaling factors $\alpha_q/\sum_{r=1}^{N_q}\alpha_r$ for a one-dimensional fifth-order finite element containing a discontinuity at its center. The uniform scaling (blue) uses $\alpha_q=1$. The WENO scaling (red) uses the weights $\alpha_q$ given by \ref{['eq:alphaq']}.
• Figure 3: Numerical solutions to the solid body rotation problem leveque1996 at $t=1$ obtained using $N_h=257^2$ and $p\in\{1,2,4\}$.
• Figure 4: Numerical solutions to the 1D inviscid Burgers equation at $t=0.2$ obtained using $N_h=128$ and $p\in\{1,2,4\}$.
• Figure 5: Numerical solutions to the KPP problem kurganov2007 at $t=1$ obtained using $N_h=257^2$ and $p=2$.

Theorems & Definitions (8)

• remark 1
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• remark 3
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• remark 8