Hamilton-Jacobi Equations

Abstract

In this chapter we discuss numerical methods for the solution of general Hamilton-Jacobi equations of the form ϕt+H(∇ϕ)=0ϕt+H(∇ϕ)=0{\phi _t} + H\left( {abla \phi } \right) = 0 (5.1) where H can be a function of both space and time. In three spatial dimensions, we can write ϕt+H(ϕx,ϕy,ϕz)=0ϕt+H(ϕx,ϕy,ϕz)=0{\phi _t} + H\left( {{\phi _x},{\phi _y},{\phi _z}} \right) = 0 (5.2) as an expanded version of equation (5.1). Convection in an externally generated velocity field (equation (3.2)) is an example of a Hamilton-Jacobi equation where H(∇φ) = 0056;↦ ·∇φ. The level set equation (equation (4.4)) is another example of a Hamilton-Jacobi equation with H(∇φ) = V n |∇φ| Here V n can depend on 0078;↦, t, or even ∇φ /|∇φ|.