| Issue |
ESAIM: M2AN
Volume 57, Number 4, July-August 2023
|
|
|---|---|---|
| Page(s) | 2493 - 2528 | |
| DOI | https://doi.org/10.1051/m2an/2023059 | |
| Published online | 01 August 2023 | |
Error estimates of a theta-scheme for second-order mean field games
1
Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratoire des signaux et systèmes, 91190 Gif-sur-Yvette, France
2
Institut Polytechnique de Paris, CNRS, Ecole Polytechnique, CMAP, 91120 Palaiseau, France
* Corresponding author: kang.liu@polytechnique.edu
Received:
15
December
2022
Accepted:
25
June
2023
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 a coupled system of the Fokker–Planck and the Hamilton–Jacobi–Bellman equation. The theta-method is used for discretizing the diffusion terms: we approximate them with a convex combination of an implicit and an explicit term. On contrast, we use an explicit centered scheme for the first-order terms. Assuming that the running cost is strongly convex and regular, we first prove the monotonicity and the stability of our thetascheme, under a CFL condition. Taking advantage of the regularity of the solution of the continuous problem, we estimate the consistency error of the theta-scheme. Our main result is a convergence rate of order O(hr) for the theta-scheme, where ℎ is the step length of the space variable and r ∈ (0, 1) is related to the Hölder continuity of the solution of the continuous problem and some of its derivatives.
Mathematics Subject Classification: 65M06 / 65M012 / 9108 / 91A16 / 49N80
Key words: Mean field games / numerical method / finite-difference scheme / error estimates
© The authors. Published by EDP Sciences, SMAI 2023
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