Yong-Tao Zhang

High Order Fast Sweeping Methods for Static Hamilton---Jacobi Equations

We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton–Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximations to derivatives, monotone numerical Hamiltonians and Gauss–Seidel iterations with alternating-direction sweepings. Based on well-developed first order sweeping methods, we design a novel approach to incorporate high order approximations to derivatives into numerical Hamiltonians such that the resulting numerical schemes are formally high order accurate and inherit the fast convergence from the alternating sweeping strategy. Extensive numerical examples verify efficiency, convergence and high order accuracy of the new methods.

Resolution of high order WENO schemes for complicated flow structures

In this short note we address the issue of numerical resolution and efficiency of high order weighted essentially nonoscillatory (WENO) schemes for computing solutions containing both discontinuities and complex solution features, through two representative numerical examples: the double Mach reflection problem and the Rayleigh-Taylor instability problem. We conclude that for such solutions with both discontinuities and complex solution features, it is more economical in CPU time to use higher order WENO schemes to obtain comparable numerical resolution.

Fast Sweeping Methods for Eikonal Equations on Triangular Meshes

The original fast sweeping method, which is an efficient iterative method for stationary Hamilton-Jacobi equations, relies on natural ordering provided by a rectangular mesh. We propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh. To that end we introduce multiple reference points and order all the nodes according to their

A Fast Sweeping Method for Static Convex Hamilton-Jacobi Equations

l^p

Efficient semi-implicit schemes for stiff systems

-metrics to those reference points. We show that these orderings satisfy the two most important properties underlying the fast sweeping method: (1) these orderings can cover all directions of information propagating efficiently; (2) any characteristic can be decomposed into a finite number of pieces and each piece can be covered by one of the orderings. We prove the convergence of the new algorithm. The computational complexity of the algorithm is nearly optimal in the sense that the total computational cost consists of

Bare Bones Pattern Formation: A Core Regulatory Network in Varying Geometries Reproduces Major Features of Vertebrate Limb Development and Evolution

O(M)

Multistage interaction of a shock wave and a strong vortex

flops for iteration steps and

Compact integration factor methods in high spatial dimensions

O(M{\rm log}M)

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

flops for sorting at the predetermined initialization step which can be efficiently optimized by adopting a linear time sorting method, where

Operator splitting implicit integration factor methods for stiff reaction-diffusion-advection systems

M