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High-Order Oscillation-Eliminating Hermite WENO Method for Hyperbolic Conservation Laws

Chuan Fan, Kailiang Wu

TL;DR

The paper addresses spurious oscillations in high-order HWENO schemes for hyperbolic conservation laws. It introduces an oscillation-eliminating (OE) procedure applied after each Runge–Kutta stage, based on a damping equation with an exact, discretization-free solution that damps the first-order moments via a filter, while a dimensionless transformation yields a scale- and evolution-invariant HWENO reconstruction $\mathcal{H}_{D}$ and an OE operator $\mathcal{F}_{OE}$. The main contributions include an efficient, non-intrusive OE filter that preserves high-order accuracy and spectral properties, a bound-preserving analysis via optimal convex decomposition ($OCAD$) in both 1D and 2D, and a 2D extension with reduced decomposition costs. Numerical tests demonstrate accuracy, resolution, and robustness under a normal CFL, without problem-dependent parameters.

Abstract

This paper proposes high-order accurate, oscillation-eliminating Hermite weighted essentially non-oscillatory (OE-HWENO) finite volume schemes for hyperbolic conservation laws. The OE-HWENO schemes apply an OE procedure after each Runge--Kutta stage, dampening the first-order moments of the HWENO solution to suppress spurious oscillations without any problem-dependent parameters. This OE procedure acts as a filter, derived from the solution operator of a novel damping equation, solved exactly without discretization. As a result, the OE-HWENO method remains stable with a normal CFL number, even for strong shocks producing highly stiff damping terms. To ensure the method's non-oscillatory property across varying scales and wave speeds, we design a scale- and evolution-invariant damping equation and propose a dimensionless transformation for HWENO reconstruction. The OE-HWENO method offers several advantages over existing HWENO methods: the OE procedure is efficient and easy to implement, requiring only simple multiplication of first-order moments; it preserves high-order accuracy, local compactness, and spectral properties. The non-intrusive OE procedure can be integrated seamlessly into existing HWENO codes. Finally, we analyze the bound-preserving (BP) property using optimal cell average decomposition, relaxing the BP time step-size constraint and reducing decomposition points, improving efficiency. Extensive benchmarks validate the method's accuracy, efficiency, resolution, and robustness.

High-Order Oscillation-Eliminating Hermite WENO Method for Hyperbolic Conservation Laws

TL;DR

The paper addresses spurious oscillations in high-order HWENO schemes for hyperbolic conservation laws. It introduces an oscillation-eliminating (OE) procedure applied after each Runge–Kutta stage, based on a damping equation with an exact, discretization-free solution that damps the first-order moments via a filter, while a dimensionless transformation yields a scale- and evolution-invariant HWENO reconstruction and an OE operator . The main contributions include an efficient, non-intrusive OE filter that preserves high-order accuracy and spectral properties, a bound-preserving analysis via optimal convex decomposition () in both 1D and 2D, and a 2D extension with reduced decomposition costs. Numerical tests demonstrate accuracy, resolution, and robustness under a normal CFL, without problem-dependent parameters.

Abstract

This paper proposes high-order accurate, oscillation-eliminating Hermite weighted essentially non-oscillatory (OE-HWENO) finite volume schemes for hyperbolic conservation laws. The OE-HWENO schemes apply an OE procedure after each Runge--Kutta stage, dampening the first-order moments of the HWENO solution to suppress spurious oscillations without any problem-dependent parameters. This OE procedure acts as a filter, derived from the solution operator of a novel damping equation, solved exactly without discretization. As a result, the OE-HWENO method remains stable with a normal CFL number, even for strong shocks producing highly stiff damping terms. To ensure the method's non-oscillatory property across varying scales and wave speeds, we design a scale- and evolution-invariant damping equation and propose a dimensionless transformation for HWENO reconstruction. The OE-HWENO method offers several advantages over existing HWENO methods: the OE procedure is efficient and easy to implement, requiring only simple multiplication of first-order moments; it preserves high-order accuracy, local compactness, and spectral properties. The non-intrusive OE procedure can be integrated seamlessly into existing HWENO codes. Finally, we analyze the bound-preserving (BP) property using optimal cell average decomposition, relaxing the BP time step-size constraint and reducing decomposition points, improving efficiency. Extensive benchmarks validate the method's accuracy, efficiency, resolution, and robustness.
Paper Structure (22 sections, 11 theorems, 150 equations, 13 figures, 4 tables)

This paper contains 22 sections, 11 theorems, 150 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Numerical schemes
  3. Framework of OE-HWENO approach
  4. One-dimensional OE-HWENO method
  5. HWENO operator $\mathcal{H}$
  6. OE operator $\mathcal{F}_{\rm OE}$
  7. Extension to 1D hyperbolic systems
  8. Scale Invariance
  9. Evolution invariance
  10. Analysis of approximate dispersion relation (ADR)
  11. Two-dimensional OE-HWENO method
  12. HWENO Operator $\mathcal{H}$
  13. OE operator $\mathcal{F}$
  14. Extension to 2D hyperbolic systems
  15. Bound Preservation via Optimal Convex Decomposition
  16. ...and 7 more sections

Key Result

Theorem 1

Denote $(\bar{u}_i^{\sigma},\bar{v}_i^{\sigma}):=\bm{U}^{\sigma}_i$. The OE procedure $\bm{U}^{\sigma}_i=\mathcal{F}_{\rm OE}\{\bm{U}_{j}\}_{j\in\Lambda_i}$ can be exactly solved and explicitly expressed as where the coefficient $\widehat{\sigma}_{i}(u_h^*)$ is defined in sec3:sigma_1d. For the 1D sixth-order OE-HWENO scheme, the jumps $[\![\partial^m_x u_h^*]\!]_{i+\frac{1}{2}}$ in sec3:sigma_1d

Figures (13)

  • Figure 2.1: The dispersion and dissipation errors of HWENO and OE-HWENO schemes.
  • Figure 3.1: Numerical results computed by the OE-HWENO and OF-HWENO methods with 800 uniform cells for the equations \ref{['sec3:LWR_Eq1']}, \ref{['sec3:LWR_Eq2']}, and \ref{['sec3:LWR_Eq3']} (from left to right) for Example \ref{['sec3:Example_LWR']}.
  • Figure 3.2: Density of Lax problem computed by OE-HWENO and OF-HWENO schemes with $200$ uniform cells.
  • Figure 3.3: Density of the Woodward-Colella blast wave problem computed by the OE-HWENO and OF-HWENO methods with $800$ uniform cells.
  • Figure 3.4: Numerical results of the 1D Sedov blast wave problem computed by the St-HWENO (top), OF-HWENO (middle) and OE-HWENO (bottom) methods with $800$ uniform cells.
  • ...and 8 more figures

Theorems & Definitions (41)

  • Theorem 1: Exact solver of OE procedure
  • proof
  • Remark 1: Stability
  • Remark 2: Conservation
  • Remark 3: Efficiency and Simplicity
  • Theorem 2: Maintain accuracy
  • proof
  • Theorem 3
  • Definition 1: Scale invariance
  • Theorem 4
  • ...and 31 more