  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.
