A high-order finite difference HWENO scheme with WENO-η non-linear weights for Hamilton-Jacobi equations

We introduce a novel family of Hermite weighted essentially non-oscillatory (HWENO) solvers based on the finite difference framework for Hamilton-Jacobi (HJ) equations. The temporal discretization advances not only the solution itself but also its first spatial derivatives. A novel WENO-η non-linear weighting procedure is introduced to replace the classical WENO-Z weights. The WENO-η weights incorporate a problem-dependent parameter η that controls the sensitivity to smoothness indicators, leading to improved resolution near derivative discontinuities while maintaining high-order accuracy in smooth regions. Extensive one- and two-dimensional numerical experiments, including linear advection, Burgers type equations, non-convex Hamiltonians, and problems from optimal control and level set methods, demonstrate that the new HWENO-η schemes achieve superior resolution compared to classical HWENO-C and HWENO-NC schemes, with essentially non-oscillatory behavior.