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A moment-based Hermite WENO scheme with unified stencils for hyperbolic conservation laws

Chuan Fan, Jianxian Qiu, Zhuang Zhao

TL;DR

The paper tackles high-order, non-oscillatory solutions for hyperbolic conservation laws by introducing a fifth-order HWENO-U scheme that evolves zeroth- and first-order moments with unified stencils and applies a HWENO limiter during time discretization. A key contribution is the scale-invariant nonlinear weighting based on integral averages $u_{ave}$, which improves robustness across sharp scale variations and ensures identical weights for $u$ and $\zeta u$. The stability analysis via Fourier methods shows that modifying the first-order moment in time discretization is essential for stability, yielding a CFL-like bound of $0<\Delta t/\Delta x\lesssim 0.824$; extensive one- and two-dimensional tests demonstrate fifth-order accuracy, high resolution, and robustness against shocks, outperforming competing HWENO-M and WENO-ZQ schemes in efficiency at similar costs. The approach simplifies implementation with a single reconstruction on unified stencils and is readily extendable to unstructured meshes, broadening applicability to complex flows.

Abstract

In this paper, a fifth-order moment-based Hermite weighted essentially non-oscillatory scheme with unified stencils (termed as HWENO-U) is proposed for hyperbolic conservation laws. The main idea of the HWENO-U scheme is to modify the first-order moment by a HWENO limiter only in the time discretizations using the same information of spatial reconstructions, in which the limiter not only overcomes spurious oscillations well, but also ensures the stability of the fully-discrete scheme. For the HWENO reconstructions, a new scale-invariant nonlinear weight is designed by incorporating only the integral average values of the solution, which keeps all properties of the original one while is more robust for simulating challenging problems with sharp scale variations. Compared with previous HWENO schemes, the advantages of the HWENO-U scheme are: (1) a simpler implemented process involving only a single HWENO reconstruction applied throughout the entire procedures without any modifications for the governing equations; (2) increased efficiency by utilizing the same candidate stencils, reconstructed polynomials, and linear and nonlinear weights in both the HWENO limiter and spatial reconstructions; (3) reduced problem-specific dependencies and improved rationality, as the nonlinear weights are identical for the function $u$ and its non-zero multiple $ζu$. Besides, the proposed scheme retains the advantages of previous HWENO schemes, including compact reconstructed stencils and the utilization of artificial linear weights. Extensive benchmarks are carried out to validate the accuracy, efficiency, resolution, and robustness of the proposed scheme.

A moment-based Hermite WENO scheme with unified stencils for hyperbolic conservation laws

TL;DR

The paper tackles high-order, non-oscillatory solutions for hyperbolic conservation laws by introducing a fifth-order HWENO-U scheme that evolves zeroth- and first-order moments with unified stencils and applies a HWENO limiter during time discretization. A key contribution is the scale-invariant nonlinear weighting based on integral averages , which improves robustness across sharp scale variations and ensures identical weights for and . The stability analysis via Fourier methods shows that modifying the first-order moment in time discretization is essential for stability, yielding a CFL-like bound of ; extensive one- and two-dimensional tests demonstrate fifth-order accuracy, high resolution, and robustness against shocks, outperforming competing HWENO-M and WENO-ZQ schemes in efficiency at similar costs. The approach simplifies implementation with a single reconstruction on unified stencils and is readily extendable to unstructured meshes, broadening applicability to complex flows.

Abstract

In this paper, a fifth-order moment-based Hermite weighted essentially non-oscillatory scheme with unified stencils (termed as HWENO-U) is proposed for hyperbolic conservation laws. The main idea of the HWENO-U scheme is to modify the first-order moment by a HWENO limiter only in the time discretizations using the same information of spatial reconstructions, in which the limiter not only overcomes spurious oscillations well, but also ensures the stability of the fully-discrete scheme. For the HWENO reconstructions, a new scale-invariant nonlinear weight is designed by incorporating only the integral average values of the solution, which keeps all properties of the original one while is more robust for simulating challenging problems with sharp scale variations. Compared with previous HWENO schemes, the advantages of the HWENO-U scheme are: (1) a simpler implemented process involving only a single HWENO reconstruction applied throughout the entire procedures without any modifications for the governing equations; (2) increased efficiency by utilizing the same candidate stencils, reconstructed polynomials, and linear and nonlinear weights in both the HWENO limiter and spatial reconstructions; (3) reduced problem-specific dependencies and improved rationality, as the nonlinear weights are identical for the function and its non-zero multiple . Besides, the proposed scheme retains the advantages of previous HWENO schemes, including compact reconstructed stencils and the utilization of artificial linear weights. Extensive benchmarks are carried out to validate the accuracy, efficiency, resolution, and robustness of the proposed scheme.
Paper Structure (10 sections, 2 theorems, 59 equations, 16 figures, 6 tables)

This paper contains 10 sections, 2 theorems, 59 equations, 16 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Description of HWENO-U scheme
  3. One-dimensional case
  4. Two-dimensional case
  5. Stability analysis
  6. Numerical tests
  7. Accuracy tests
  8. Non-smooth tests
  9. Concluding remarks
  10. Appendix

Key Result

Theorem 2.1

Combining the semi-discrete HWENO scheme sec3:1dlinear_HWENO with either the stand forward Euler or third-order SSP RK time discretization s2, the resulting schemes are both unstable. Proof. The semi-discrete HWENO scheme sec3:1dlinear_HWENO with the forward Euler time method is To apply the von-Neumann analysis, we have an assumption on the solution where $\sigma$ is the imaginary unit satisfy

Figures (16)

  • Figure 2.1: The big stencil $S_0$, small stencils $\{S_m\}_{m=1}^4$ and their respective labels.
  • Figure 2.2: The polar plot of the eigenvalues and amplification matrix.
  • Figure 3.1: Comparison of $L^\infty$, $L^1$ errors and CPU time for Example \ref{['Example:Burgers1DTestOrder']}.
  • Figure 3.2: Comparison of $L^\infty$, $L^1$ errors and CPU time for Example \ref{['Example:Euler1DTestOrder']}.
  • Figure 3.3: Comparison of $L^\infty$, $L^1$ errors and CPU time for Example \ref{['Example:Burgers2DTestOrder']}.
  • ...and 11 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • ...and 7 more