Sparse grid implementation of a fixed-point fast sweeping WENO scheme for Eikonal equations

TL;DR

This work addresses the high computational cost of high-order fixed-point fast sweeping WENO methods for multidimensional Eikonal (static Hamilton-Jacobi) equations. It couples a third-order WENO spatial discretization with a second-order Runge-Kutta fixed-point fast sweeping framework and then accelerates the computation via the sparse-grid combination technique, using third-order WENO prolongation. Across 2D and 3D tests—both smooth and non-smooth—the sparse-grid RK FPFS-WENO method achieves substantial CPU-time savings (typically 50–90%) while maintaining comparable accuracy and resolution to full-grid runs. The results support the practical utility of sparse grids for high-order nonlinear HJ problems, though theoretical error analysis for nonlinear sparse-grid schemes remains an open area for future work.

Abstract

Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs). As other types of fast sweeping schemes, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order. The resulting iterative schemes have fast convergence rate to steady state solutions. Moreover, an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve inverse operation of any nonlinear local system. Hence they are robust and flexible, and have been combined with high order accurate weighted essentially non-oscillatory (WENO) schemes to solve various hyperbolic PDEs in the literature. For multidimensional nonlinear problems, high order fixed-point fast sweeping WENO methods still require quite large amount of computational costs. In this technical note, we apply sparse-grid techniques, an effective approximation tool for multidimensional problems, to fixed-point fast sweeping WENO method for reducing its computational costs. Here we focus on a robust Runge-Kutta (RK) type fixed-point fast sweeping WENO scheme with third order accuracy (Zhang et al. 2006 [33]), for solving Eikonal equations, an important class of static Hamilton-Jacobi (H-J) equations. Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse grid computations of the fixed-point fast sweeping WENO scheme achieve large savings of CPU times on refined meshes, and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.

Figures (8)

• Figure 1: Stencils of the third-order WENO approximations for derivatives.
• Figure 2: Illustration of 2D sparse grids $\{\Omega^{l_1,l_2}\}$ for one cell of a root grid. Here the cell indicated by the levels $l_1=0, l_2=0$ is one cell of the root grid $\Omega^{0,0}$, and the side length of the cell is $H$. The finest level $N_L=3$. Highlighted grids are those on which PDEs are solved.
• Figure 3: Example 3, numerical solutions of the two-sphere problem by the RK FPFS-WENO scheme on sparse grids ($N_r = 80$ for root grid, finest level $N_L=3$ in the sparse-grid computation) and the corresponding $640\times640\times640$ single grid, using the third order WENO interpolation for prolongation in the sparse-grid combination. (a), (c), (e): single-grid result; (b), (d), (f): sparse-grid result; (a), (b): the contour plots for $\phi=0.5$; (c), (d): the contour plots for $\phi=1$; (e), (f): the contour plots for the whole surface.
• Figure 4: Example 4, numerical solutions of the shape-from-shading problem by the RK FPFS-WENO scheme on sparse grids ($N_r = 160$ for root grid, finest level $N_L=3$ in the sparse-grid computation) and the corresponding $1280\times1280$ single grid, using the third order WENO interpolation for prolongation in the sparse-grid combination. (a), (b): single-grid result; (c), (d): sparse-grid result; (a), (c): three-dimensional view of the solutions; (b), (d): the contour plots, 30 equally spaced contour lines from $\phi = 0$ to $\phi = 2$.
• Figure 5: Example 5, Case 1, numerical solutions of the 2D Voronoi diagram problem by the RK FPFS-WENO scheme on sparse grids ($N_r = 160$ for root grid, finest level $N_L=3$ in the sparse-grid computation) and the corresponding $1280\times1280$ single grid, using the third order WENO interpolation for prolongation in the sparse-grid combination. The contour plots, 30 equally spaced contour lines from $\phi = 0$ to $\phi = 0.5589$. Red points are the generators. (a): single-grid result; (b): sparse-grid result.