Discretization and convergence of the ballistic Benamou-Brenier formulation of the porous medium and Burgers equations
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917062311411712 |
|---|---|
| author | Mirebeau, Jean-Marie Stampfli, Erwan |
| author_facet | Mirebeau, Jean-Marie Stampfli, Erwan |
| contents | We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as forward in time variants of the Benamou-Brenier formulation of optimal transport. This approach turns those evolution problems into global optimization problems in time and space, of which we introduce a discretization, one of whose originalities lies in the harmonic interpolation of the densities involved. We prove that the resulting schemes are unconditionally stable w.r.t. the space and time steps, and we establish a quadratic convergence rate for the dual PDE solution, under suitable assumptions. We also show that the schemes can be efficiently solved numerically using a proximal splitting method and a global space-time fast Fourier transform, and we illustrate our results with numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02662 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discretization and convergence of the ballistic Benamou-Brenier formulation of the porous medium and Burgers equations Mirebeau, Jean-Marie Stampfli, Erwan Numerical Analysis 65M06 (Primary) 65M12, 49M29 (Secondary) We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as forward in time variants of the Benamou-Brenier formulation of optimal transport. This approach turns those evolution problems into global optimization problems in time and space, of which we introduce a discretization, one of whose originalities lies in the harmonic interpolation of the densities involved. We prove that the resulting schemes are unconditionally stable w.r.t. the space and time steps, and we establish a quadratic convergence rate for the dual PDE solution, under suitable assumptions. We also show that the schemes can be efficiently solved numerically using a proximal splitting method and a global space-time fast Fourier transform, and we illustrate our results with numerical experiments. |
| title | Discretization and convergence of the ballistic Benamou-Brenier formulation of the porous medium and Burgers equations |
| topic | Numerical Analysis 65M06 (Primary) 65M12, 49M29 (Secondary) |
| url | https://arxiv.org/abs/2511.02662 |