Dissipation-based WENO stabilization of high-order finite element methods for scalar conservation laws

TL;DR

This paper introduces a new WENO-based nonlinear stabilization technique for high-order finite element methods solving scalar conservation laws, blending high- and low-order stabilization to improve shock handling while maintaining accuracy.

Contribution

It proposes a novel WENO stabilization approach using a smoothness sensor and Hermite interpolation, offering an alternative to traditional WENO limiters in high-order finite element methods.

Findings

The method effectively captures shocks with low-order diffusion.

The scheme achieves optimal convergence rates on general meshes.

Numerical experiments confirm the theoretical error estimates.

Abstract

We present a new perspective on the use of weighted essentially nonoscillatory (WENO) reconstructions in high-order methods for scalar hyperbolic conservation laws. The main focus of this work is on nonlinear stabilization of continuous Galerkin (CG) approximations. The proposed methodology also provides an interesting alternative to WENO-based limiters for discontinuous Galerkin (DG) methods. Unlike Runge--Kutta DG schemes that overwrite finite element solutions with WENO reconstructions, our approach uses a reconstruction-based smoothness sensor to blend the numerical viscosity operators of high- and low-order stabilization terms. The so-defined WENO approximation introduces low-order nonlinear diffusion in the vicinity of shocks, while preserving the high-order accuracy of a linearly stable baseline discretization in regions where the exact solution is sufficiently smooth. The underlying reconstruction procedure performs Hermite interpolation on stencils consisting of a mesh cell and its neighbors. The amount of numerical dissipation depends on the relative differences between partial derivatives of reconstructed candidate polynomials and those of the underlying finite element approximation. All derivatives are taken into account by the employed smoothness sensor. To assess the accuracy of our CG-WENO scheme, we derive error estimates and perform numerical experiments. In particular, we prove that the consistency error of the nonlinear stabilization is of the order $p+1/2$, where $p$ is the polynomial degree. This estimate is optimal for general meshes. For uniform meshes and smooth exact solutions, the experimentally observed rate of convergence is as high as $p+1$.