Strongly consistent low-dissipation WENO schemes for finite elements

TL;DR

This paper introduces a novel WENO stabilization method for finite element discretizations that maintains strong consistency, improves accuracy, and effectively captures shocks and steep gradients in conservation laws.

Contribution

It presents a new blending approach for WENO schemes using residual-based weights, enhancing accuracy and stability in finite element methods for conservation laws.

Findings

Achieves improved accuracy over weakly consistent WENO schemes.

Demonstrates stability and effectiveness in hyperbolic conservation laws.

Numerical results outperform traditional WENO methods in shock scenarios.

Abstract

We propose a way to maintain strong consistency and facilitate error analysis in the context of dissipation-based WENO stabilization for continuous and discontinuous Galerkin discretizations of conservation laws. Following Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) and Vedral (arXiv preprint arXiv:2309.12019), we use WENO shock detectors to determine appropriate amounts of low-order artificial viscosity. In contrast to existing WENO methods, our approach blends candidate polynomials using residual-based nonlinear weights. The shock-capturing terms of our stabilized Galerkin methods vanish if residuals do. This enables us to achieve improved accuracy compared to weakly consistent alternatives. As we show in the context of steady convection-diffusion-reaction (CDR) equations, nonlinear local projection stabilization terms can be included in a way that preserves the coercivity of local bilinear forms. For the corresponding Galerkin-WENO discretization of a CDR problem, we rigorously derive a priori error estimates. Additionally, we demonstrate the stability and accuracy of the proposed method through one- and two-dimensional numerical experiments for hyperbolic conservation laws and systems thereof. The numerical results for representative test problems are superior to those obtained with traditional WENO schemes, particularly in scenarios involving shocks and steep gradients.