Error estimates of a theta-scheme for second-order mean field games

We introduce and analyze a new finite-difference scheme, relying on the theta-method, for solving monotone second-order mean field games. These games consist of coupled system Fokker–Planck Hamilton–Jacobi–Bellman equation. The theta-method is used discretizing diffusion terms: we approximate them with convex combination an implicit explicit term. On contrast, use centered scheme first-order terms. Assuming that running cost strongly regular, first prove monotonicity stability our thetascheme, under CFL condition. Taking advantage regularity solution continuous problem, estimate consistency error theta-scheme. Our main result convergence rate order O ( h r ) theta-scheme, where ℎ step length space variable ∈ (0, 1) related to Hölder continuity problem some its derivatives.