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The jump filter in the discontinuous Galerkin method for hyperbolic conservation laws

Lei Wei, Lingling Zhou, Yinhua Xia

TL;DR

The paper tackles spurious oscillations in high-order DG discretizations of hyperbolic conservation laws by introducing a jump-based local viscosity that acts as a shock-capturing filter. Implemented via a time-splitting strategy, the filter damps high-order polynomial modes based on intercell jumps while preserving conservation and $L^2$ stability, with proven optimal convergence for smooth solutions. The method extends naturally to 2D and systems, and can be combined with a hybrid limiter to further reduce dissipation without compromising accuracy or compactness. Numerical experiments on 1D and 2D Euler equations validate effective oscillation control, sharp shock resolution, and compatibility with efficient, parallelizable computations, highlighting practical impact for robust, high-order DG simulations.

Abstract

When simulating hyperbolic conservation laws with discontinuous solutions, high-order linear numerical schemes often produce undesirable spurious oscillations. In this paper, we propose a jump filter within the discontinuous Galerkin (DG) method to mitigate these oscillations. This filter operates locally based on jump information at cell interfaces, targeting high-order polynomial modes within each cell. Besides its localized nature, our proposed filter preserves key attributes of the DG method, including conservation, $L^2$ stability, and high-order accuracy. We also explore its compatibility with other damping techniques, and demonstrate its seamless integration into a hybrid limiter. In scenarios featuring strong shock waves, this hybrid approach, incorporating this jump filter as the low-order limiter, effectively suppresses numerical oscillations while exhibiting low numerical dissipation. Additionally, the proposed jump filter maintains the compactness of the DG scheme, which greatly aids in efficient parallel computing. Moreover, it boasts an impressively low computational cost, given that no characteristic decomposition is required and all computations are confined to physical space. Numerical experiments validate the effectiveness and performance of our proposed scheme, confirming its accuracy and shock-capturing capabilities.

The jump filter in the discontinuous Galerkin method for hyperbolic conservation laws

TL;DR

The paper tackles spurious oscillations in high-order DG discretizations of hyperbolic conservation laws by introducing a jump-based local viscosity that acts as a shock-capturing filter. Implemented via a time-splitting strategy, the filter damps high-order polynomial modes based on intercell jumps while preserving conservation and stability, with proven optimal convergence for smooth solutions. The method extends naturally to 2D and systems, and can be combined with a hybrid limiter to further reduce dissipation without compromising accuracy or compactness. Numerical experiments on 1D and 2D Euler equations validate effective oscillation control, sharp shock resolution, and compatibility with efficient, parallelizable computations, highlighting practical impact for robust, high-order DG simulations.

Abstract

When simulating hyperbolic conservation laws with discontinuous solutions, high-order linear numerical schemes often produce undesirable spurious oscillations. In this paper, we propose a jump filter within the discontinuous Galerkin (DG) method to mitigate these oscillations. This filter operates locally based on jump information at cell interfaces, targeting high-order polynomial modes within each cell. Besides its localized nature, our proposed filter preserves key attributes of the DG method, including conservation, stability, and high-order accuracy. We also explore its compatibility with other damping techniques, and demonstrate its seamless integration into a hybrid limiter. In scenarios featuring strong shock waves, this hybrid approach, incorporating this jump filter as the low-order limiter, effectively suppresses numerical oscillations while exhibiting low numerical dissipation. Additionally, the proposed jump filter maintains the compactness of the DG scheme, which greatly aids in efficient parallel computing. Moreover, it boasts an impressively low computational cost, given that no characteristic decomposition is required and all computations are confined to physical space. Numerical experiments validate the effectiveness and performance of our proposed scheme, confirming its accuracy and shock-capturing capabilities.
Paper Structure (10 sections, 2 theorems, 59 equations, 12 figures, 2 tables, 1 algorithm)

This paper contains 10 sections, 2 theorems, 59 equations, 12 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Scheme formulation
  3. One-dimensional scalar conservation laws
  4. Stability and error estimates
  5. Two-dimensional scalar conservation laws
  6. Systems of conservation laws
  7. Numerical experiments
  8. One dimensional tests
  9. Two dimensional tests
  10. Conclusions

Key Result

Theorem 2.1

Suppose that without the jump filter, the fully discrete RKDG scheme is stable under the CFL condition $\tau\leq\gamma h^{\kappa}$, where $\kappa=1+1/r$ for $r\equiv 1$ (mod $4$), $\kappa=1+1/(r+1)$ for $r\equiv 2$ (mod $4$), and $\kappa=1$, $C_{*}=0$ for $r\equiv3$ (mod $4$). Let $u_{\sigma}^{n}$ b under the same CFL condition $\tau \leq \gamma h^\kappa$. Here $C$ is a positive constant independe

Figures (12)

  • Figure 3.1: Numerical results of DG scheme with the jump filter for Example \ref{['exm:burgers']}, with $N=120$ and $P^k$, $k=1,2,3$, at time $T=1.5$. Top: numerical solutions of full polynomials and cell averages. Bottom: pointwise errors and $\sigma_{j,l}$ where $l=k$.
  • Figure 3.2: Numerical results of DG scheme with the jump filter and hybrid limiters. Left: Example \ref{['ex:Lax']} , with $N=200$ and $P^2$, at time $T= 0.13$. Right: Example \ref{['ex:Blast']} , with $N=600$ and $P^2$, at time $T= 0.038$.
  • Figure 3.3: Numerical results of DG scheme with the jump filter and hybrid limiters for Example \ref{['ex:ShuOsher']} with $N=400$ at time $T= 1.8$. From left to right: $P^1$; $P^2$; $P^1$ (Zoomed-in); $P^2$ (Zoomed-in).
  • Figure 3.4: Numerical results of DG scheme with the jump filter for Example \ref{['ex:2Dvortex']} case (b). From left to right, and top to bottom: $T = 0.068$; $T = 0.203$; $T = 0.330$; $T = 0.529$; $T = 0.662$; $T = 0.8$. 30 equally spaced pressure contours from 0.68 to 1.30 at $T = 0.068$ and $T = 0.203$; $T = 0.330$. 90 equally spaced pressure contours from 1.19 to 1.37 at $T = 0.529$; $T = 0.662$; and $T = 0.8$. $P^3$ basis. $400\times 200$ cells.
  • Figure 3.5: Numerical results for Example \ref{['ex:Double_rec']}. DG solutions of $k=1,2,3,4$ from top to bottom. Left: the jump filter; Right: the hybrid limiter associated with the jump filter. 30 equally spaced density contours from $1.5$ to $21.5$. $960 \times 240$ cells.
  • ...and 7 more figures

Theorems & Definitions (13)

  • Theorem 2.1
  • Theorem 2.2
  • Example 3.1: Burgers equation
  • Example 3.2: Lax problem
  • Example 3.3: Blast waves
  • Example 3.4: Shu-Osher shock tube problem
  • Example 3.5: Vortex evolution problems
  • Example 3.6: Double Mach reflection problem
  • Example 3.7: Forward facing step problem
  • Example 3.8: Shock passing a backward facing corner
  • ...and 3 more