High Order Finite Difference Hermite WENO Fast Sweeping Methods for Static Hamilton-Jacobi Equations (Journal of Computational Mathematics, 2023)
In this paper, we propose a novel Hermite weighted essentially non-oscillatory (HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently. During theHWENO reconstruction procedure, the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils. However,one major novelty and difference from the traditional HWENO framework lies in the factthat, we do not need to introduce and solve any additional equations to update the derivatives of the unknown function $φ$. Instead, we use the current $φ$ and the old spatial derivativeof $φ$ to update them. The traditional HWENO fast sweeping method is also introduced inthis paper for comparison, where additional equations governing the spatial derivatives of $φ$ are introduced. The novel HWENO fast sweeping methods are shown to yield great savings in computational time, which improves the computational efficiency of the traditionalHWENO scheme. In addition, a hybrid strategy is also introduced to further reduce computational costs. Extensive numerical experiments are provided to validate the accuracyand efficiency of the proposed approaches.
Keywords: Finite difference; Hermite methods; Weighted essentially non-oscillatory method; Fast sweeping method; Static Hamilton-Jacobi equations; Eikonal equation.