Overview
This paper introduces high-order weighted essentially non-oscillatory (WENO)-based semi-implicit Newton-type Gauss-Seidel Lax-Friedrichs fast sweeping methods designed for solving generalized Eikonal equations. These equations are relevant to wave propagation through moving fluids. The proposed methods build upon a pre-existing Newton-type framework, enhancing its local solver to achieve higher orders of accuracy.
Research Context
The research is situated within the domain of numerical methods for solving static Hamilton-Jacobi equations, specifically the generalized Eikonal equation. The foundation for this work is a Newton-type framework developed by Li and Qian (2020). This prior framework involved updating the solution line-wise and utilized Newton's method with a tridiagonal and strictly diagonally dominant Jacobian.
Approach
The core of the proposed approach involves extending the local solver of the Li and Qian (2020) Newton-type framework. This extension is achieved by integrating high-order WENO approximations for the spatial derivatives into the numerical Hamiltonian. This modification allows the new methods to reach fifth-, seventh-, and ninth-order accuracy.
Within the Gauss-Seidel iteration framework, three distinct alternating sweeping strategies are employed:
- Column-wise sweeping
- Row-wise sweeping
- Column-row-wise sweeping
Findings
Numerical examples were conducted to evaluate the proposed schemes. These examples spanned both two and three spatial dimensions. The results from these numerical examples demonstrated both the efficiency and accuracy of the introduced high-order WENO-based semi-implicit Newton-type fast sweeping methods. The specific performance metrics or quantitative comparisons are not detailed in the abstract.