Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods
The authors study a class of semi-Lagrangian schemes for the Hamilton-Jacobi equation of the first order, in particular, they consider the model problem: \[ v_{t}(t)+H(\nabla v(t))=0, x\in \mathbb{R}^{N}, t\in \mathbb{R}, v(x,0)=v_{0}(x), x\in \mathbb{R}^{N}. \] These schemes can be interpreted as a discrete analogue of the Hopf-Lax-Oleinik representation formula for the exact viscosity solution of the first order evaluative Hamilton-Jacobi equations. The authors study the semi-Lagrangian schemes for the Hamilton-Jacobi equation as an analogue of Godunov scheme for conservation laws and show that these schemes are strictly connected with the Hopf representation formulae for the exact solution. In the end of the paper they analyse the numerical experiments with tests for the eikonal propagation of fronts, for the strictly convex Hamiltonian and for a front evolving on a manifold.
scientific article; zbMATH DE number 1763713
scientific article