New Finite Difference Hermite WENO Schemes for Hamilton–Jacobi Equations

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.