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Oscillation-eliminating central DG schemes for hyperbolic conservation laws

Manting Peng, Kailiang Wu, Caiyou Yuan

TL;DR

The paper addresses spurious oscillations in central discontinuous Galerkin methods for hyperbolic conservation laws by introducing the oscillation-eliminating CDG (OECDG) framework, which integrates a novel dual damping mechanism with an OE procedure based on overlapping CDG solutions. It establishes rigorous theoretical foundations, proving approximate skew-symmetry and weak boundedness of the CDG discretization, proving linear stability of RKCDG via a matrix-transfer approach, and deriving fully discrete optimal error estimates for OECDG even in nonlinear damping scenarios. The method is demonstrated on a broad suite of 1D and 2D problems, including linear and nonlinear advection, Burgers, traffic flow, nonconvex fluxes, and Euler equations, showing accurate convergence, sharp resolution of complex wave patterns, and effective suppression of nonphysical oscillations without heavy stencils or characteristic decompositions. Overall, OECDG offers a robust, high-order, compact, and parameter-free approach for oscillation control in hyperbolic systems, with strong theoretical guarantees and wide practical applicability.

Abstract

This paper proposes and analyzes a class of essentially non-oscillatory central discontinuous Galerkin (CDG) methods for general hyperbolic conservation laws. First, we introduce a novel compact, non-oscillatory stabilization mechanism that effectively suppresses spurious oscillations while preserving the high-order accuracy of CDG methods. Unlike existing limiter-based approaches that rely on large stencils or problem-specific parameters for oscillation control, our dual damping mechanism is inspired by CDG-based numerical dissipation and leverages overlapping solutions within the CDG framework, significantly enhancing stability while maintaining compactness. Our approach is free of problem-dependent parameters and complex characteristic decomposition, making it efficient and robust. Second, we provide a rigorous stability and optimal error analysis for fully discrete Runge-Kutta (RK) CDG schemes, addressing a gap in the theoretical understanding of these methods. Specifically, we establish the approximate skew-symmetry and weak boundedness of the CDG discretization. These results enable us to rigorously analyze the fully discrete error estimates for our oscillation-eliminating CDG (OECDG) method, a challenging task due to its nonlinear nature, even for linear advection equations. Building on this framework, we reformulate nonlinear oscillation-eliminating CDG schemes as linear RK CDG schemes with a nonlinear source term, extending error estimates beyond the linear case to schemes with nonlinear oscillation control. While existing error analyses for DG or CDG schemes have largely been restricted to linear cases without nonlinear oscillation-control techniques, our analysis represents an important theoretical advancement. Experiments validate the theoretical findings and demonstrate the effectiveness of the OECDG method.

Oscillation-eliminating central DG schemes for hyperbolic conservation laws

TL;DR

The paper addresses spurious oscillations in central discontinuous Galerkin methods for hyperbolic conservation laws by introducing the oscillation-eliminating CDG (OECDG) framework, which integrates a novel dual damping mechanism with an OE procedure based on overlapping CDG solutions. It establishes rigorous theoretical foundations, proving approximate skew-symmetry and weak boundedness of the CDG discretization, proving linear stability of RKCDG via a matrix-transfer approach, and deriving fully discrete optimal error estimates for OECDG even in nonlinear damping scenarios. The method is demonstrated on a broad suite of 1D and 2D problems, including linear and nonlinear advection, Burgers, traffic flow, nonconvex fluxes, and Euler equations, showing accurate convergence, sharp resolution of complex wave patterns, and effective suppression of nonphysical oscillations without heavy stencils or characteristic decompositions. Overall, OECDG offers a robust, high-order, compact, and parameter-free approach for oscillation control in hyperbolic systems, with strong theoretical guarantees and wide practical applicability.

Abstract

This paper proposes and analyzes a class of essentially non-oscillatory central discontinuous Galerkin (CDG) methods for general hyperbolic conservation laws. First, we introduce a novel compact, non-oscillatory stabilization mechanism that effectively suppresses spurious oscillations while preserving the high-order accuracy of CDG methods. Unlike existing limiter-based approaches that rely on large stencils or problem-specific parameters for oscillation control, our dual damping mechanism is inspired by CDG-based numerical dissipation and leverages overlapping solutions within the CDG framework, significantly enhancing stability while maintaining compactness. Our approach is free of problem-dependent parameters and complex characteristic decomposition, making it efficient and robust. Second, we provide a rigorous stability and optimal error analysis for fully discrete Runge-Kutta (RK) CDG schemes, addressing a gap in the theoretical understanding of these methods. Specifically, we establish the approximate skew-symmetry and weak boundedness of the CDG discretization. These results enable us to rigorously analyze the fully discrete error estimates for our oscillation-eliminating CDG (OECDG) method, a challenging task due to its nonlinear nature, even for linear advection equations. Building on this framework, we reformulate nonlinear oscillation-eliminating CDG schemes as linear RK CDG schemes with a nonlinear source term, extending error estimates beyond the linear case to schemes with nonlinear oscillation control. While existing error analyses for DG or CDG schemes have largely been restricted to linear cases without nonlinear oscillation-control techniques, our analysis represents an important theoretical advancement. Experiments validate the theoretical findings and demonstrate the effectiveness of the OECDG method.

Paper Structure

This paper contains 17 sections, 11 theorems, 85 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. OECDG method
  3. Semi-discrete CDG formulation
  4. Fully-discrete OECDG methods
  5. Theoretical analysis
  6. Properties of CDG discretization
  7. Linear stability of RKCDG method
  8. Optimal error estimates of fully-discrete OECDG method
  9. Numerical examples
  10. 1D linear advection equation
  11. 1D inviscid Burgers equation
  12. Lighthill--Whitham--Richards traffic flow model lu2008explicit
  13. 1D compressible Euler equations
  14. 2D linear advection equation
  15. A nonconvex nonlinear conservation law
  16. ...and 2 more sections

Key Result

Lemma 3.1

\newlabellem:skew-symm0 The CDG discretization operator $\mathcal{H}(\boldsymbol{h},\boldsymbol{g})$ is approximately skew-symmetric:

Figures (10)

  • Figure 1: Results of the $\mathbb{P}^1$, $\mathbb{P}^2$, and $\mathbb{P}^3$ OECDG methods (from left to right) at $t=1.1$.
  • Figure 2: OECDG solutions for 1D Burgers equation. Form left to right: $\mathbb{P}^1$, $\mathbb{P}^2$, and $\mathbb{P}^3$.
  • Figure 3: Density profiles of traffic flow at $t = 0.5h$, $1.0h$, and $1.5h$ (left to right).
  • Figure 4: Density at $t = 1.8$ computed by $\mathbb{P}^1$, $\mathbb{P}^2$, and $\mathbb{P}^3$ OECDG (from left to right).
  • Figure 5: Densities of OECDG solutions for \ref{['Sod']} to \ref{['leblanc']}.
  • ...and 5 more figures

Theorems & Definitions (43)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Proof 1
  • Remark 3.2
  • Lemma 3.3
  • Proof 2
  • Definition 3.4
  • Theorem 3.5
  • Proof 3
  • ...and 33 more