New finite difference Hermite WENO schemes for Hamilton-Jacobi equations (Q2307427)

In the literature, there exist various high-order accurate essentially non-oscillatory (HWENO) schemes for solving the Hamilton-Jacobi equations. In this paper, a new Hermite weighted essentially non-oscillatory (HWENO) schemes is developed for solving the Hamilton-Jacobi equations in one and two dimensions on structured meshes. The new HWENO schemes are constructed using Hermite interpolation with a series of unequal-sized spatial stencils and a third-order TVD Runge-Kutta time discretization method. In the classical HWENO schemes, only the HWENO reconstructions are used. The new HWENO schemes are designed by evolving the solution and its first derivatives and using them in the HWENO reconstructions and high-order linear reconstructions. The new HWENO schemes are simple, effective, and converge fast to non-viscosity solutions compared with the other existing HWENO schemes. In smooth regions, smaller truncation errors are obtained in all accuracy test cases in comparison to the existing HWENO schemes. Extensive benchmark examples are performed including linear and nonlinear Burger's equation, Riemann problem with a nonconvex flux, Eikonal equation in one and two dimensions, to illustrate the good performance of such new HWENO schemes.