Abstract

In this paper, new finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are designed for solving the Hamilton–Jacobi equations on structured meshes. The crucial idea of the spatial reconstructions is borrowed from the original HWENO schemes (Qiu and Shu in J Comput Phys 204:82–99, 2005), in which the function and its first derivative values are evolved in time and used in the reconstruction. Such new HWENO spatial reconstructions with the application of three unequal-sized spatial stencils result in an important innovation that we perform only spatial HWENO reconstructions for numerical fluxes of function values and high-order linear reconstructions for numerical fluxes of derivatives, which are different to other HWENO schemes. The new HWENO schemes could obtain smaller errors with optimal high-order accuracy in smooth regions, and keep sharp transitions and non-oscillatory property near discontinuities. Extensive benchmark examples are performed to illustrate the good performance of such new finite difference HWENO schemes.

中文翻译：

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Hamilton-Jacobi方程的新的有限差分Hermite WENO方案

摘要

在本文中，设计了新的有限差分Hermite加权基本非振荡（HWENO）方案，用于求解结构化网格上的Hamilton-Jacobi方程。空间重构的关键思想是从原始的HWENO方案中借用的（Qiu和Shu，J Comput Phys 204：82–99，2005），其中功能及其一阶导数会随时间演化并用于重构。这种使用三个不等大小的空间模具的新HWENO空间重构导致了一项重要的创新，即我们仅对函数值的数值通量执行空间HWENO重构，对导数的数值通量执行高阶线性重构，这与其他方法不同HWENO方案。新的HWENO方案可以在平滑区域中以最佳的高阶精度获得较小的误差，并在不连续点附近保持尖锐的过渡和非振荡特性。进行了大量的基准测试示例，以说明这种新的有限差分HWENO方案的良好性能。