HIGH-ORDER FACTORIZATION BASED HIGH-ORDER HYBRID FAST SWEEPING METHODS FOR POINT-SOURCE EIKONAL EQUATIONS

The solution for the eikonal equation with a point-source condition has an upwind singularity at the source point as the eikonal solution behaves like a distance function at and near the source. As such, the eikonal function is not differentiable at the source so that all formally high-order numerical schemes for the eikonal equation yield first-order convergence and relatively large errors. Therefore, it is a longstanding challenge in computational geometrical optics how to compute a uniformly high-order accurate solution for the point-source eikonal equation in a global domain. In this paper, assuming that both the squared slowness and the squared eikonal are analytic near the source, we propose high-order factorization based high-order hybrid fast sweeping methods for point-source eikonal equations to compute just such solutions. Observing that the squared eikonal is differentiable at the source, we propose to factorize the eikonal into two multiplicative or additive factors, one of which is specified to approximate the eikonal up to arbitrary order of accuracy near the source, and the other of which serves as a higher-order correction term. This decomposition is achieved by using the eikonal equation and applying power series expansions to both the squared eikonal and the squared slowness function. We develop recursive formulas to compute the approximate eikonal up to arbitrary order of accuracy near the source. Furthermore, these approximations enable us to decompose the eikonal into two factors, either multiplicatively or additively, so that we can design two new types of hybrid, high-order fast sweeping schemes for the point-source eikonal equation. We also show that the first-order hybrid fast sweeping methods are monotone and consistent so that they are convergent in computing viscosity solutions. Two-and three-dimensional numerical examples demonstrate that a hybrid pth order fast sweeping method yields desired, uniform, clean pth order convergence in a global domain by using a pth order factorization.