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In \cite{christlieb2019kernel}, the authors developed a class of high-ordernumerical schemes for the Hamilton-Jacobi (H-J) equations, which areunconditionally stable, yet take the form of an explicit scheme. This paperextends such schemes, so that they are more effective at capturing sharpgradients, especially on nonuniform meshes. In particular, we modify theweighted essentially non-oscillatory (WENO) methodology in the previouslydeveloped schemes by incorporating an exponential basis and adapting thepreviously developed nonlinear filters used to control oscillations. The mainadvantages of the proposed schemes are their effectiveness and simplicity,since they can be easily implemented on higher-dimensional nonuniform meshes.We perform numerical experiments on a collection of examples, including H-Jequations with linear, nonlinear, convex and non-convex Hamiltonians. Todemonstrate the flexibility of the proposed schemes, we also include testproblems defined on non-trivial geometry.

A Kernel-Based Explicit Unconditionally Stable Scheme for Hamilton-Jacobi Equations on Nonuniform Meshes