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Strongly consistent low-dissipation WENO schemes for finite elements

Joshua Vedral, Andreas Rupp, Dmitri Kuzmin

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.

Strongly consistent low-dissipation WENO schemes for finite elements

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.
Paper Structure (19 sections, 8 theorems, 83 equations, 5 figures)

This paper contains 19 sections, 8 theorems, 83 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Stabilized Galerkin methods
  3. Dissipation-based nonlinear stabilization
  4. Residual-based WENO scheme
  5. A priori analysis for CG-WENO methods
  6. Problem statement
  7. Stability
  8. Existence of solutions
  9. Preliminaries
  10. A priori estimates
  11. A priori analysis for DG-WENO methods
  12. Numerical examples
  13. Solid body rotation
  14. KPP problem
  15. Euler equations of gas dynamics
  16. ...and 4 more sections

Key Result

Lemma 5.1

The stabilized scheme eq:wfs is coercive, i.e., and it is bounded in the sense that there is a constant $\hat{C}>0$ such that

Figures (5)

  • Figure 1: Solid body rotation, numerical solutions at $t=1.0$ obtained using $E_h=128^2$ and $p=2$.
  • Figure 2: KPP problem, numerical solutions at $t=1.0$ obtained using $E_h=128^2$ and $p=2$.
  • Figure 3: Titarev-Toro problem, density profiles $\varrho$ at $t=5.0$ obtained using $E_h=1000$ and $p\in\{1,2\}$.
  • Figure 4: Kelvin-Helmholtz instability, density profiles $\varrho$ at $t=1.0$ obtained using $E_h=512^2$ and $p=1$.
  • Figure 5: Double Mach reflection, density profiles $\varrho$ at $t=0.2$ obtained using $E_h=768\cdot192$ and $p=2$.

Theorems & Definitions (21)

  • Remark 3.1
  • Remark 3.2
  • Remark 4.1
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Theorem 5.3
  • proof
  • Remark 5.4
  • ...and 11 more