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

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.