Hermite WENO schemes for Hamilton–Jacobi equations on unstructured meshes

In this paper, we extend a class of the Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the Hamilton–Jacobi equations by Qiu and Shu (2005) [24] to two dimensional unstructured meshes. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first two derivative values are evolved via time advancing and used in the reconstructions, while only the function values are evolved and used in the original WENO schemes which are nodal based approximations. The third and fourth order HWENO schemes using the combinations of second order approximations with nonlinear weights and TVD Runge–Kutta time discretization method are used here. Comparing with the original WENO schemes for Hamilton–Jacobi equations, one major advantage of HWENO schemes presented here is its compactness in the reconstructions. Extensive numerical tests are performed to illustrate the capability and high order accuracy of the methodologies.