Abstract
In this article, we propose and study several discrete versions of homogeneous and inhomogeneous one-dimensional Fokker–Planck equations. In particular, for these discretizations of velocity and space, we prove the exponential convergence to the equilibrium of the solutions, for time-continuous equations as well as for time-discrete equations. Our method uses new types of discrete Poincaré inequalities for a “two-direction” discretization of the derivative in velocity. For the inhomogeneous problem, we adapt hypocoercive methods to the discrete cases.
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Notes
i.e. involving the space variable x and the velocity variable v.
i.e. involving the variable v but not the variable x.
Note that, in these definitions, the range of indices of the image \(\mathsf {D}_vG\) is \(\mathbb Z^*\) and not \(\mathbb Z\), in order to keep into account the natural shift induced by the “two-direction” definition of \(\mathsf {D}_v\).
We emphasize the fact that there is no mistake in the denominator of \((\mathsf {D}^\sharp _vG)_0\).
Once again, there is no typo in the formula defining \((\mathsf {D}^\sharp _vH)_0\).
Note anyway that the explicit Euler scheme
$$\begin{aligned} f^{n+1} = f^n -{\delta \! t}( - \mathsf {D}^\sharp _v+{v^\sharp }) \mathsf {D}_vf^n , \end{aligned}$$is not well posed due to the fact that the discretized operator \(( - \mathsf {D}^\sharp _v+{v^\sharp }) \mathsf {D}_v\) is not bounded.
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Guillaume Dujardin is supported by the Inria project-team MEPHYSTO and the Labex CEMPI (ANR-11-LABX-0007-01). Frédéric Hérau is supported by the Grant “NOSEVOL” ANR-2011-BS01019-01 and the Labex Centre Henri Lebesgue (ANR-11-LABX-0020-01).
Appendix: Commutation identities
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Dujardin, G., Hérau, F. & Lafitte, P. Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker–Planck equations. Numer. Math. 144, 615–697 (2020). https://doi.org/10.1007/s00211-019-01094-y
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DOI: https://doi.org/10.1007/s00211-019-01094-y