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

TL;DR

This paper introduces a high-order, oscillation-eliminating Hermite WENO scheme for hyperbolic conservation laws that effectively suppresses spurious oscillations, maintains stability at high CFL numbers, and preserves accuracy and efficiency.

Contribution

The paper develops a novel OE procedure integrated into HWENO schemes, acting as a filter derived from a damping equation, enhancing stability and non-oscillatory behavior without problem-dependent parameters.

Findings

The OE-HWENO method achieves high-order accuracy and stability for problems with strong shocks.

The method effectively suppresses spurious oscillations without sacrificing resolution.

Extensive benchmarks demonstrate improved efficiency, robustness, and resolution over existing schemes.

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.