A Method to Solve Hamilton–Jacobi Type Equation on Unstructured Meshes

Abstract

A new method is developed to approximate a first-order Hamilton–Jacobi equation. The constant motion of an interface in the normal direction is of interest. The interface is captured with the help of a “Level-Set” function approximated through a finite-volume Godunov-type scheme. Contrarily to most computational approaches that consider smooth Level-Set functions, the present one considers sharp “Level-Set”, the numerical diffusion being controlled with the help of the Overbee limiter (Chiapolino et al. in J Comput Phys 340:389–417, 2017). The method requires gradient computation that is addressed through the least squares approximation. Multidimensional results on fixed unstructured meshes are provided and checked against analytical solutions. Geometrical properties such as interfacial area and volume computation are addressed as well. Results show excellent agreement with the exact solutions.