# A hybrid Hermite WENO scheme for hyperbolic conservation laws

## TL;DR

This paper introduces a hybrid HWENO scheme combining shock detection and adaptive reconstruction for hyperbolic conservation laws, improving accuracy and efficiency near discontinuities while maintaining robustness and compactness.

## Contribution

The paper presents a novel hybrid HWENO scheme that adaptively switches between high-order linear and HWENO reconstructions based on shock detection, enhancing performance over existing methods.

## Key findings

- Demonstrates high accuracy and resolution in benchmark tests.
- Shows robustness and efficiency in handling discontinuities.
- Maintains scheme compactness and reduces oscillations.

## Abstract

In this paper, we propose a hybrid finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for solving one and two dimensional hyperbolic conservation laws. The zeroth-order and the first-order moments are used in the spatial reconstruction, with total variation diminishing Runge-Kutta time discretization. The main idea of the hybrid HWENO scheme is that we first use a shock-detection technique to identify the troubled cell, then, if the cell is identified as a troubled cell, we would modify the first order moment in the troubled cell and employ HWENO reconstruction in spatial discretization; otherwise, we directly use high order linear reconstruction. Unlike other HWENO schemes, we borrow the thought of limiter for discontinuous Galerkin (DG) method to control the spurious oscillations, after this procedure, the scheme would avoid the oscillations by using HWENO reconstruction nearby discontinuities and have higher efficiency for using linear approximation straightforwardly in the smooth regions. In addition, the hybrid HWENO scheme still keeps the compactness. A collection of benchmark numerical tests for one and two dimensional cases are performed to demonstrate the numerical accuracy, high resolution and robustness of the proposed scheme.