The Numerical Method For Hamilton-Jacobi Equations On Triangular Meshes

Hamilton-Jacobi (H-J) equations are of important application in many fields, such as geometric optics, computational fluid dynamics, control systems, computer vision, and mesh generation. For this reason, in the last decade, there is much theoretical and numerical study on H-J equations. Generally it is very difficult to find the analytic solutions of H-J equations and their weak solutions are not unique. Even if Hamiltonian H and initial condition u 0 are continuous, they may develop discontinuous derivatives. About the construction of numerical methods in solving H-J equations on unstructured meshes, the difficulty is mainly in the choice of numerical flux and the construction of the high-order interpolation polynomials. In 1996, Abgrall gave a numerical flux of H-J equations and the scheme is only first-order. Therefore, the construction of high-order accuracy schemes is of great importance.In this paper we obtained a high-order scheme for H-J equations by constructing the high-order interpolation polynomials on triangular meshes. The detailed procedure is as follows: constructing interpolation polynomials on every triangular cell by solving linear equation systems. If equations are ill conditioned, they can be solved with the least square method by adding more vertexes gradually into the interpolating stencil. The interpolation polynomials are more accurate as more information on the neighboring triangular cells is considered. The obtained schemes are employed to simulate some typical examples; it is found that the constructed schemes in this paper are of high-order accuracy in smooth regions and good resolution.