High order finite difference Hermite WENO fixed-point fast sweeping method for static Hamilton-Jacobi equations

TL;DR

This paper introduces an efficient HWENO fixed-point fast sweeping method for static Hamilton-Jacobi equations, improving computational efficiency, accuracy, and simplicity over traditional schemes through novel reconstruction and hybrid strategies.

Contribution

The paper develops a new HWENO framework that avoids auxiliary equations, uses fixed-point fast sweeping for efficiency, and introduces a hybrid strategy to further reduce computational costs.

Findings

The proposed method reduces computational time compared to traditional HWENO schemes.

It achieves higher accuracy with smaller numerical errors on the same mesh.

The hybrid strategy further decreases computational cost without sacrificing performance.

Abstract

In this paper, we combine the nonlinear HWENO reconstruction in \cite{newhwenozq} and the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static Hamilton-Jacobi equations in a novel HWENO framework recently developed in \cite{mehweno1}. The proposed HWENO frameworks enjoys several advantages. First, compared with the traditional HWENO framework, the proposed methods do not need to introduce additional auxiliary equations to update the derivatives of the unknown function $\phi$. They are now computed from the current value of $\phi$ and the previous spatial derivatives of $\phi$. This approach saves the computational storage and CPU time, which greatly improves the computational efficiency of the traditional HWENO scheme. In addition, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping method is used to update the numerical approximation. It is an explicit method and does not involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves some known issues, including that the iterative number is very sensitive to the parameter $\varepsilon$ used in the nonlinear weights, as observed in previous studies. Finally, in order to further reduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate the good performance of the proposed fixed-point fast sweeping HWENO methods.