Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.22016 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913152030998528 |
|---|---|
| author | Cui, Jianbo Dang, Tonghe |
| author_facet | Cui, Jianbo Dang, Tonghe |
| contents | We prove first-order convergence of semi-discrete monotone finite difference schemes for Hamilton--Jacobi equations on the Wasserstein space over a finite graph. A central challenge is the boundary degeneracy of the Wasserstein simplex, which prevents the direct use of the standard $L^1$ adjoint method and limits doubling-of-variables arguments to the suboptimal rate $\mathcal O(h^{\frac 12})$ \cite{CDM25}. We address this issue by introducing a weighted $L^1$ framework with a boundary-vanishing weight and by analyzing the corresponding weighted adjoint equation for the linearized operator of the scheme, featuring a new geometric drift term. Our proof relies on uniform bounds for the weighted adjoint variable and the mesh-parameter derivative of the numerical solution. These estimates are derived from discrete gradient and semi-concavity bounds, obtained through a bootstrap argument for two classes of monotone Hamiltonians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22016 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | First-Order Convergence of Monotone Schemes for Hamilton--Jacobi Equations on the Wasserstein Space on Graphs Cui, Jianbo Dang, Tonghe Numerical Analysis 49L25, 65M15, 35R02 We prove first-order convergence of semi-discrete monotone finite difference schemes for Hamilton--Jacobi equations on the Wasserstein space over a finite graph. A central challenge is the boundary degeneracy of the Wasserstein simplex, which prevents the direct use of the standard $L^1$ adjoint method and limits doubling-of-variables arguments to the suboptimal rate $\mathcal O(h^{\frac 12})$ \cite{CDM25}. We address this issue by introducing a weighted $L^1$ framework with a boundary-vanishing weight and by analyzing the corresponding weighted adjoint equation for the linearized operator of the scheme, featuring a new geometric drift term. Our proof relies on uniform bounds for the weighted adjoint variable and the mesh-parameter derivative of the numerical solution. These estimates are derived from discrete gradient and semi-concavity bounds, obtained through a bootstrap argument for two classes of monotone Hamiltonians. |
| title | First-Order Convergence of Monotone Schemes for Hamilton--Jacobi Equations on the Wasserstein Space on Graphs |
| topic | Numerical Analysis 49L25, 65M15, 35R02 |
| url | https://arxiv.org/abs/2605.22016 |