High-order WENO-based semi-implicit Newton-type fast sweeping methods for static Hamilton-Jacobi equations
Abstract
In this paper, we propose high-order weighted essentially non-oscillatory (WENO)-based semi-implicit Newton-type Gauss-Seidel Lax-Friedrichs fast sweeping methods for solving the generalized Eikonal equation arising in wave propagation through a moving fluid. Building upon the Newton-type framework of Li and Qian (2020), which updates the solution line-wise using Newton's method with a tridiagonal and strictly diagonally dominant Jacobian, we extend the local solver to fifth-, seventh-, and ninth-order accuracy by incorporating high-order WENO approximations of the spatial derivatives into the numerical Hamiltonian. Three alternating sweeping strategies, namely column-wise, row-wise, and column-row-wise, are considered within the Gauss-Seidel iteration framework. Numerical examples in both two and three spatial dimensions demonstrate the efficiency and accuracy of the proposed schemes.
Cite
@article{arxiv.2608.01304,
title = {High-order WENO-based semi-implicit Newton-type fast sweeping methods for static Hamilton-Jacobi equations},
author = {Yuan Liu and Jianliang Qian},
journal= {arXiv preprint arXiv:2608.01304},
year = {2026}
}