In Zhu (2014) , we presented a new type of the finite volume Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the Hamilton-Jacobi equations on structured meshes, the key idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however the cell averages of the function and its two derivative values are together evolved via time approaching and used in the reconstructions, while only the function is evolved and used in the original WENO schemes which are nodal based approximations on finite difference framework. In this paper, we extend the method to solve the Hamilton-Jacobi equations on unstructured meshes. The major advantages of the new HWENO schemes on unstructured meshes are their compactness in the spatial field, purely on the finite volume framework and only one set of small stencils is needed for different types of the polynomial reconstruction. Extensive numerical tests are performed to illustrate the capability and high order of accuracy of these methodologies.
Zhu, J., & Qiu, J. (2014). Finite volume Hermite WENO schemes for solving the Hamilton-Jacobi equations II: Unstructured meshes. Computers and Mathematics with Applications, 68(10), 1137–1150. https://doi.org/10.1016/j.camwa.2014.08.013