Alternating Evolution Method Sovling Static Hamilton-Jacobi Equations

In this work, we propose a novel approach to solve nonlinear static HamiltonJacobi equations. In order to overcome the challenge of the nonlinear HJ equation, as well as computing the viscosity solution, we design alternating evolution schemes, which are based on the alternating evolution reformulation of the original Hamilton-Jacobi equation, and then reconstruct polynomials to approximate the exact solution. In this article, the structure of the iterative formula creates an arti-?cial parameter varepsilon, and the selection of the parameters affects the stability and convergence of the iteration. So in the third chapter, we give the stability and convergence analysis of one-dimensional problems for the ?rst order AE scheme,and stability analysis for the second order AE scheme, as well as stability analysis of ?rst order AE scheme for two-dimensional problems. We select typical numerical example to verify both the accuracy and simplicity of the AE method in solving the steady Hamilton- Jacobi equation.We also apply the proposed method to the kinetic Hamilton-Jacobi equation, in which the Hamiltonian is determined by an integral in phase space.