SOLUTION OF STEADY-STATE HAMILTON-JACOBI EQUATION BASED ON ALTERNATING EVOLUTION METHOD

Abstract

Hamilton-Jacobi equation is a kind of highly nonlinear partial differential equation which is difficult to be solved.  The boundary value problem of steady-state Hamilton-Jacobi equation is supposed as H(x,∇xφ(x)) = 0, x ∈ Ω/Γ; φ(x) = q(x), x ∈ Γ (Ω ∈ Rd,d stands for the space dimensionality, Ω stands for a bounded open set with a boundary of Γ, and H stands for a given non-linear function, called Hamiltonian).  Even though Hamiltonian function is smooth, the derivative of its solution may be disconnected at some cuspidal points. There are many ways to solve a steady-state Hamilton-Jacobi equation, among which, fast marching method (FMM) and fast sweeping method (FSM) are famous.  This study solved Hamilton-Jacobi equation using alternating evolution method (AE).  Firstly, an initial Hamilton-Jacobi equation was described using AE; then polynomials were constructed to approach the Hamilton-Jacobi equation and the equation was finally solved by selecting proper iterative methods and correct boundary conditions.  An artificial parameter was generated in the process of construction of iterative format; the selection of the parameter could directly affect the stability and convergence of the iterative format.  On account of this, the stability and convergence of the first-order AE algorithm was analyzed and the effectiveness and accuracy of the algorithm was proved by a numerical experiment.