Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods

We study a class of semi-Lagrangian schemes which can be interpreted as a discrete version of the Hopf-Lax-Oleinik representation formula for the exact viscosity solution of first order evolutive Hamilton-Jacobi equations. That interpretation shows that the scheme is potentially accurate to any prescribed order. We discuss how the method can be implemented for convex and coercive hamiltonians with a particular structure and how this method can be coupled with a discrete Legendre trasform. We also show that in one dimension the first order semi-Lagrangian scheme coincides with the integration of the Godunov scheme for the corresponding conservation laws. Several test illustrate the main features of semi-Lagrangian schemes for evolutive Hamilton-Jacobi equations.