On numerical approximation of the Hamilton-Jacobi-transport system arising in high frequency approximations

Abstract

22 pagesInternational audienceIn the present article, we study the numerical approximation of a system of Hamilton-Jacobi and transport equations arising in geometrical optics. We consider a semi-Lagrangian scheme. We prove the well posedness of the discrete problem and the convergence of the approximated solution toward the viscosity-measure valued solution of the exact problem

A semi-Lagrangian scheme for a degenerate second order mean field game system

“…(ii) All the results of this paper, can be extended for the more general Hamiltonians H(x, t, p) considered in [1]. In fact, consider the system…”

“…The second result is the discrete semiconcavity of v ε ρ,h [µ] (see e.g. [1]), which implies a.e. convergence of Dv ε ρ,h [µ] to Dv[µ] (where v[µ] is the unique viscosity solution of (1.2)).…”

A Fully Discrete Semi-Lagrangian Scheme for a First Order Mean Field Game Problem

“…This fact completely changes the theoretical and numerical analysis of the problem. As a matter of fact, for example in [3], the key property for convergence result of the proposed numerical scheme is a one side Lipschitz condition for Dv(·, ·) of the form:(1.3)By the results in [27], condition (1.3) assures the stability of the so-called Fillipov characteristics and of the associated measure solutions of the continuity equation, which are the key to obtain their convergence result. Unfortunately, in our case (1.3) corresponds to the semiconvexity of v, which does not holds for an arbitrary time horizon T (see [11]).…”

“…This fact completely changes the theoretical and numerical analysis of the problem. As a matter of fact, for example in [3], the key property for convergence result of the proposed numerical scheme is a one side Lipschitz condition for Dv(·, ·) of the form:…”

On the Discretization of Some Nonlinear Fokker--Planck--Kolmogorov Equations and Applications

“…Damped oscillator: On the left we display the contour level sets for m ρ,h (x, t) = 0.2 at times t = 0.2, 0.5, 1 and 2. The black point corresponds to (1,1), which is the point where the initial mass is concentrated. On the right, we display a 3D view of the numerical solution at time T = 2 computed with ρ = 0.025.…”