We construct Two-Point Flux Approximation (TPFA) finite volume schemes to solve the quadratic optimal transport problem in its dynamic form, namely the problem originally introduced by Benamou and Brenier. We show numerically that these type of discretizations are prone to form instabilities in their more natural implementation, and we propose a variation based on nested meshes in order to overcome these issues. Despite the lack of strict convexity of the problem, we also derive quantitative estimates on the convergence of the method, at least for the discrete potential and the discrete cost. Finally, we introduce a strategy based on the barrier method to solve the discrete optimization problem.
The volumetric stretch energy has been widely applied to the computation of volume-/mass-preserving parameterizations of simply connected tetrahedral mesh models. However, this approach still lacks theoretical support. In this paper, we provide the theoretical foundation for volumetric stretch energy minimization (VSEM) to compute volume-/mass-preserving parameterizations. In addition, we develop an associated efficient VSEM algorithm with guaranteed asymptotic R-linear convergence. Furthermore, based on the VSEM algorithm, we propose a projected gradient method for the computation of the volu…
In this paper, we consider a drift-diffusion charge transport model for perovskite solar cells, where electrons and holes may diffuse linearly (Boltzmann approximation) or nonlinearly (e.g. due to Fermi-Dirac statistics). To incorporate volume exclusion effects, we rely on the Fermi-Dirac integral of order -1 when modeling moving anionic vacancies within the perovskite layer which is sandwiched between electron and hole transport layers. After non-dimensionalization, we first prove a continuous entropy-dissipation inequality for the model. Then, we formulate a corresponding two-point flux fini…
The aim of this work is to construct efficient finite volume schemes for the numerical study of sediment transport in shallow water, in the framework of the Exner model.In most cases, the velocity related to the sediment is much lower that the fluid velocity, which, in turn, may be much lower that the free-surface wave speed. Explicit methods that resolve all waves require small time steps due to the CFL stability restriction because of fast surface waves. Furthermore, if Rusanov flux is adopted, slow sediment waves may be affected by the large numerical diffusion. The objective of the present…
An initial-boundary value problem describing moisture transport in porous media: existence of strong solutions and an error estimate for a finite volume scheme
Convergence of a Finite-Volume Scheme for a Degenerate Cross-Diffusion Model for Ion Transport
Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé（法国国家信息与自动化研究所，里尔大学，法国国家科学研究中心，联合研究实验室8524-保罗·潘勒瓦实验室）；Univ. Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，联合研究实验室8524，法国国家信息与自动化研究所-保罗·潘勒瓦实验室）；Institute for Analysis and Scientific Computing, Vienna University of Technology（维也纳工业大学分析与科学计算研究所）
We develop a new finite volume method using unstructured mesh-vertex grids for coupled systems modeling shallow water flows and solute transport over complex bottom topography. Novel well-balanced positivity preserving discretization techniques are proposed for the water surface elevation and the concentration of the pollutant. For the hydrodynamic system, the proposed scheme preserves the steady state of a lake at rest and the positivity of the water depth. For the scalar transport equation, the proposed method guarantees the positivity and a perfect balance of the scalar concentration. The c…
Multicellular simulations with shape and volume constraints using optimal transport
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（RIKEN跨学科理论与数学科学中心（iTHEMS））；Institute for the Advanced Study of Human Biology (ASHBi) Kyoto University（京都大学人类生物学高级研究所（ASHBi））；HeKA team, Inria Paris, Inserm, Université Paris-Cité（HeKA团队，巴黎国家信息与自动化研究所，法国国家健康与医学研究院，巴黎西岱大学）
This paper is concerned with the development of suitable numerical method for the approximation of discontinuous solutions of parameter-dependent linear hyperbolic conservation laws. The objective is to reconstruct such approximation, for new instances of the parameter values for any time, from a transformation of pre-computed snapshots of the solution trajectories for new parameter values. In a finite volume setting, a Reconstruct-Transform-Average (RTA) algorithm inspired from the Reconstruct-Evolve-Average one of Godunov's method is proposed. It allows to perform, in three steps, a transfor…
We present an efficient numerical scheme based on Monte Carlo integration to approximate statistical solutions of the incompressible Euler equations. The scheme is based on finite volume methods, which provide a more flexible framework than previously existing spectral methods for the computation of statistical solutions for incompressible flows. This finite volume scheme is rigorously proven to, under experimentally verifiable assumptions, converge in an appropriate topology and with increasing resolution to a statistical solution. The convergence obtained is stronger than that of measure-val…
Computation of volume potentials over bounded domains via approximate approximations
Sapienza University of Rome（罗马第一大学）；University of Linköping（林雪平大学）；University of Liverpool（利物浦大学）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）
Computation of Circular Area and Spherical Volume Invariants via Boundary Integrals
Department of Mathematics, University of St. Thomas（圣托马斯大学数学系）；School of Mathematics, University of Minnesota（明尼苏达大学数学学院）；Department of Anthropology, University of Minnesota（明尼苏达大学人类学系）
