Tuesday November 19th
14:00 – 15:00 Quentin Mérigot: Optimal transport and its (semi-)discretization – SLIDES
15:00 – 16:00 Claire Chainais-Hillairet: Linear / nonlinear approaches for the approximation of convection-diffusion equations – SLIDES
In this talk, I will discuss different strategies for the discretization of convection-diffusion equations by finite volume schemes. I will pay special attention to the preservation of certain physical properties by the schemes : positivity of the solutions, thermodynamic equilibria, long time behaviour.
16:30 – 17:30 Raphaèle Herbin: Staggered schemes for compressible flows – SLIDES
In the past decade, a series of numerical schemes has been developed for numerically solving fluid flow problems related to nuclear safety. These flows are inherently complex, often featuring both high and low Mach number regions. Staggered schemes, which use discrete velocity variables at the cell faces and scalar variables at the cell centers, exhibit natural stability for incompressible flows, making them suitable for discretizing such scenarios. A well-known example of a staggered scheme is the Marker and Cell (MAC) method, specifically designed for rectangular grids. This presentation will focus on these schemes applied to the compressible Euler equations. Key characteristics of these methods include the use of a staggered mesh, the discretization of the (non-conservative) internal energy balance rather than the total energy, and an upwinding of the fluxes with respect to material speed, which inherently preserves the positivity of density and internal energy. Time discretization can be achieved either explicitly or through a pressure correction technique. In both approaches, a correction term is incorporated into the discrete internal energy equation, enabling accurate shock speed recovery. Furthermore, these schemes are shown to be consistent in the Lax-Wendroff sense: under certain compactness assumptions, the sequence of approximate solutions converges to a weak solution of the Euler equations. A discrete entropy inequality is established when the mass balance and internal energy equations are discretized implicitly, with upwind fluxes.
In the case of a constant density, the semi-implicit scheme simplifies to a standard algorithm for the incompressible regime; in the case of the compressible barotropic Navier-Stokes equations. Furthermore for a fixed mesh, the numerical solution of the totally discrete compressible scheme has been proved to converge to that of the incompressible scheme as the Mach number vanishes.
17:30 – 18:30 Jürgen Fuhrmann: Install party VoronoiFVM.jl
Wednesday November 20th
9:00 – 10:00 Clément Cancès: Finite elements for W_p Wasserstein gradient flows (joint work with D. Matthes, F. Nabet and E.-M. Rott) – SLIDES
W_p Wasserstein gradient flows yield degenerate parabolic equations involving a q-Laplacian type operator, with q being p’s conjugate exponent. Specific difficulties occur when p is chosen not equal to 2. It is in particular required to compute the whole gradient of the driving potential to compute the fluxes, making usual strategies based on two-point flux approximation finite volumes irrelevant.
We propose a finite element scheme building on conformal lowest order elements with mass lumping and a backward Euler time discretization strategy. Our scheme preserves mass and positivity while energy decays in time. Its convergence can furthermore be established thanks to material from the theory of gradient flows in metric spaces by Ambrosio, Gigli & Savaré. The analytical results are confirmed by numerical simulations.
10:30 – 11:30 André Schlichting: The fourth-order DLSS equation as gradient flow of the entropy with
respect to diffusive transport and its structure preserving discretization – SLIDES
We introduce a diffusive metric between probability measures on the real line that generalizes the Hellinger, Kantorovich and martingale transport. We represent several classes of parabolic PDEs as metric gradient flows with respect to diffusive transport, as for instance linear second order diffusion equations in non-divergence form, the porous medium equation of exponent two, and the fourth order DLSS equation. We propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle, based its on the diffusive gradient flow structure. The discrete dynamics inherits this gradient flow structure, and in addition further properties, such as an alternative gradient flow formulation in the Wasserstein distance, contractivity in the Hellinger distance, and monotonicity of several Lyapunov functionals. Our main result is the convergence in the limit of vanishing mesh size. The proof
relies on a discrete version of a nonlinear functional inequality between integral expressions involving second order derivatives. We give an outlook of higher order generalizations and its convergence
properties. Moreover, we also complement the discrete scheme with an interacting particle system converging to the DLSS equation and a porous medium variant of it in the hydrodynamic limit.
Based on joints works (in progress) with Daniel Matthes, Eva-Maria Rott, Giuseppe Savaré and Artur Stephan. First result in arxiv:2312.13284.
11:30 – 12:30 Bertrand Maury: Wasserstein gradient flow interpretation of Finite Volume Schemes
The Fokker Planck equation in a euclidean domain can be interpreted as a gradient flow for the relative entropy with respect to the stationnary measure, in the Wasserstein context. At the discrete level, Jan Mass introduced in 2011 a framework which makes it possible to define a Wasserstein-like distance (in the spirit of the Benamou-Brenier approach) on a graph endowed with a Markov Kernel, and to interpret the associated Heat equation as a gradient flow associated with this distance.
With N. Gigli, he also established a few years later a discrete-to-continuous convergence result (in the Hausdorff Gromov sense) of the respective metric spaces of measures (for regular grids).
Since Finite Volume space discretizations of the continuous FP equation lead to some sort of discrete heat equations, it raises the question whether a given FV discretization strategy respects the underlying gradient flow structure. Our presentation will address this question and related ones.
14:00 – 15:00 Gabriele Todeschi: EVI flows via Nonnegative Cross-Curvature – SLIDES
15:00 – 16:00 Flore Nabet: On the finite-volume analysis of the stochastic heat equation – SLIDES
In this presentation we will focus on the finite volume approximation of the stochastic heat equation.
We will begin with a multiplicative Ito noise where we will show the convergence of the numerical scheme to the unique variational solution of the continuous problem, with particular emphasis on the stochastic compactness argument. We then turn to the problem of a Stratonovich-type transport noise and detail the advantages of this choice as well as the difficulties it causes.
This work was carried out in the one hand in collaboration with C. Bauzet, K. Schmitz and A. Zimmermann and in the other hand with A. De Bouard and L. Goudenège.
16:30 – 17:30 Lorenzo Portinale: Homogenisation of transport problems on graphs – SLIDES
We discuss discrete-to-continuum limits of optimal transport problems, with particular attention to recent contributions in the periodic and the stochastic setting. We introduce a natural discretization of a broad
class of dynamical transport problems and discuss convergence results in the setting of periodic (resp. random and stationary) graphs with periodic (resp. random) transport problems.
17:30 – 18:30 Hugo Leclerc: Install party Sdot
Sdot is an intuitive C++/Python package designed to streamline optimal transport computations in semi-discrete configurations. It comprises three main modules: a module for optimal transport operations (optimization of the potentials, transport map analysis, …), a module for power diagrams (integrations, geometric operations, …) and a module to handle convex polyhedral functions.
In this presentation, we will outline the package’s general approach and we’ll test together a few simple application cases.
Thursday November 21st
9:00 – 10:00 Oliver Tse: Generalized Gradient Structures of Finite Volume Methods – SLIDES
In this talk, I will introduce the concept of generalized gradient structures (GGS) and discuss how one can view finite volume discretization of certain PDEs as generalized gradient flows. I also plan to discuss a derivation of a GGS for the upwind scheme from a GGS of the Scharfetter-Gummel scheme.
10:30 – 11:30 Anastasiia Hraivoronska: Variational convergence for the Scharfetter-Gummel scheme – SLIDES
In this talk, we discuss the convergence of the semi-discrete Scharfetter–Gummel scheme for the aggregation-diffusion equation. We first introduce the relevant finite-volume space discretization and unravel a novel generalized gradient structure for it. We then follow a variational approach and pass to the discrete-to-continuum limit in the energy-dissipation formulation. Passing to the limit, we recover the Otto gradient flow structure for the aggregation-diffusion equation based on the 2-Wasserstein distance.
11:30 – 12:30 Virginie Ehrlacher: Results and open questions about cross-diffusion systems on moving boundary domains (joint work with Clément Cancès, Jean Cauvin-Vila and Claire Chainais-Hillairet) – SLIDES
The aim of this talk is to present some recent results and open questions about the mathematical study of two cross-diffusion systems coupled by a moving interface boundary. The motivation stems, for instance, from the modelisation of the fabrication of thin film solar cells via Solid Vapor Deposition. More precisely, the system of interest is composed of a mixture of various chemical species that belong to either a gaseous phase or a solid phase, occupying a one-dimensional model. The evolution of the interface between the two phases is driven by so-called Butler-Volmer chemical potentials. This system can be shown to have an entropic structure and a structure-preserving finite volume scheme will be presented. Several open questions about possible interesting follow-ups to this work will also be mentioned and hopefully lead to interesting discsussions with other participants of the workshop.
