Numerical discretization of the first‐order Hamilton‐Jacobi equation on triangular meshes

We present and analyze several ways of discretizing first-order Hamilton-Jacobi equations on unstructured meshes. We first discuss two Godunov-type Hamiltonians: the first one is an extension of a result by Bardi and Osher, where a particular decomposition of the initial condition is assumed, and we point out its practical limits; the other one arises from a particular decomposition of the Hamiltonian. Despite its complexity this decomposition enables us to construct a Lax-Friedrichs Hamiltonian. These schemes all share common properties: They are consistent, monotonic, and independent of the geometrical interpretation of the piecewise linear initial condition. Under these assumptions and classical ones on the mesh, we show these schemes are convergent. We describe their high-order extensions using the ENO technique and provide numerical illustrations. © 1996 John Wiley & Sons, Inc.