Newton-type Gauss–Seidel Lax–Friedrichs high-order fast sweeping methods for solving generalized eikonal equations at large-scale discretization

Abstract

We propose a Newton-type Gauss–Seidel Lax–Friedrichs sweeping method to solve the generalized eikonal equation arising from wave propagation in a moving fluid. The Lax–Friedrichs numerical Hamiltonian is used in discretization of the generalized eikonal equation. Different from traditional Lax–Friedrichs sweeping algorithms, we design a novel approach with a line-wise sweeping strategy. In the local solver, the values of traveltime on an entire line are updated simultaneously by Newton’s method. The global solution is then obtained by Gauss–Seidel iterations with line-wise sweepings. We first develop the Newton-based first-order scheme, and on top of that we further develop high-order schemes by applying weighted essentially non-oscillatory (WENO) approximations to derivatives. Extensive 2-D and 3-D numerical examples demonstrate the efficiency and accuracy of the new algorithm. The combination of Newton’s method and Gauss–Seidel iterations improves upon the convergence speed of the original Lax–Friedrichs sweeping algorithm. In addition, the Newton-type sweeping method manipulates data in a vectorized manner so that it can be efficiently implemented in modern programming languages that feature array programming, and the resulting advantages are extremely significant for large-scale 3-D computations.