Numerical Schemes for the Hamilton–Jacobi and Level Set Equations on Triangulated Domains

• Published in: Journal of computational physics 1998-09, Vol.145 (1), p.1-40
• Main Authors: Barth, Timothy J., Sethian, James A.
• Format: Article
• Language: English
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1006/jcph.1998.6007

Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the Hamilton–Jacobi (H-J), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for the H-J equations. Unfortunately, the basic scheme lacks Lipschitz continuity of the numerical Hamiltonian. By employing a “virtual” edge flipping technique, local Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local Lipschitz continuity of the numerical Hamiltonian. Finally, a class of Petrov–Galerkin approximations is considered. These schemes are stabilized via a least-squares bilinear form. The Petrov–Galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy. Discretization of the level set equation also requires the numerical approximation of a mean curvature term. A simple mass-lumped Galerkin approximation is presented in Section 6 and analyzed using maximum principle analysis. The use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples. These numerical examples include discretizations of convex and nonconvex forms of the H-J equation, the Eikonal equation, and the level set equation.