In this paper, a new type of high-order finite volume and finite difference multi-resolution Hermite weighted essentially non-oscillatory (HWENO) schemes are designed for solving hyperbolic conservation laws on structured meshes. Here we only use the information defined on a hierarchy of nested central spatial stencils but do not introduce any equivalent multi-resolution representation, the terminology of multi-resolution HWENO follows that of the multi-resolution WENO schemes (Zhu and Shu, 2018) [29]. The main idea of our spatial reconstruction is derived from the original HWENO schemes (Qiu and Shu, 2004) [19], in which both the function and its first-order derivative values are evolved in time and used in the reconstruction. Our HWENO schemes use the same large stencils as the classical HWENO schemes which are narrower than the stencils of the classical WENO schemes for the same order of accuracy. Only the function values need to be reconstructed by our HWENO schemes, the first-order derivative values are obtained from the high-order linear polynomials directly. Furthermore, the linear weights of such HWENO schemes can be any positive numbers as long as their sum equals one, and there is no need to do any modification or positivity-preserving flux limiting in our numerical experiments. Extensive benchmark examples are performed to illustrate the robustness and good performance of such finite volume and finite difference HWENO schemes.

在本文中，设计了一种新型的高阶有限体积和有限差分多分辨率 Hermite 加权本质非振荡 (HWENO) 方案，用于求解结构化网格上的双曲守恒定律。在这里，我们仅使用在嵌套中央空间模板的层次结构上定义的信息，但不引入任何等效的多分辨率表示，多分辨率 HWENO 的术语遵循多分辨率 WENO 方案的术语（Zhu 和 Shu，2018）[ 29]。我们空间重建的主要思想源自原始的 HWENO 方案（Qiu 和 Shu，2004）[19]，其中函数及其一阶导数值都随时间演化并用于重建。我们的 HWENO 方案使用与经典 HWENO 方案相同的大模板，这些模板比经典 WENO 方案的模板窄，但精度相同。我们的 HWENO 方案只需要重构函数值，直接从高阶线性多项式中获得一阶导数。此外，这种 HWENO 方案的线性权重可以是任何正数，只要它们的总和等于 1，并且在我们的数值实验中不需要做任何修改或保留正性通量限制。执行了大量基准示例以说明此类有限体积和有限差分 HWENO 方案的稳健性和良好性能。一阶导数值直接从高阶线性多项式中获得。此外，这种 HWENO 方案的线性权重可以是任何正数，只要它们的总和等于 1，并且在我们的数值实验中不需要做任何修改或保留正性通量限制。执行了大量基准示例以说明此类有限体积和有限差分 HWENO 方案的稳健性和良好性能。一阶导数值直接从高阶线性多项式中获得。此外，这种 HWENO 方案的线性权重可以是任何正数，只要它们的总和等于 1，并且在我们的数值实验中不需要做任何修改或保留正性通量限制。执行了大量基准示例以说明此类有限体积和有限差分 HWENO 方案的稳健性和良好性能。