Discretizing the Fokker-Planck equation with second-order accuracy: a dissipation driven approach
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910633193111552 |
|---|---|
| author | Cancès, Clément Monsaingeon, Léonard Natale, Andrea |
| author_facet | Cancès, Clément Monsaingeon, Léonard Natale, Andrea |
| contents | We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time discretization is novel and relies on a tailored nonlinear mid-point rule, designed to accurately capture the dissipative structure of the model. We establish well-posedness for the scheme, positivity of the solutions, as well as a fully discrete energy-dissipation inequality mimicking the continuous one. We then prove the rigorous convergence of the scheme under mildly restrictive conditions on the unstructured grids, which can be easily satisfied in practice. Numerical simulations show that our scheme is second order accurate both in time and space, and that one can solve the discrete nonlinear systems arising at each time step using Newton's method with low computational cost. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03367 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discretizing the Fokker-Planck equation with second-order accuracy: a dissipation driven approach Cancès, Clément Monsaingeon, Léonard Natale, Andrea Analysis of PDEs Numerical Analysis We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time discretization is novel and relies on a tailored nonlinear mid-point rule, designed to accurately capture the dissipative structure of the model. We establish well-posedness for the scheme, positivity of the solutions, as well as a fully discrete energy-dissipation inequality mimicking the continuous one. We then prove the rigorous convergence of the scheme under mildly restrictive conditions on the unstructured grids, which can be easily satisfied in practice. Numerical simulations show that our scheme is second order accurate both in time and space, and that one can solve the discrete nonlinear systems arising at each time step using Newton's method with low computational cost. |
| title | Discretizing the Fokker-Planck equation with second-order accuracy: a dissipation driven approach |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2410.03367 |