Dynamics of complex systems

Applied Mathematics

Institut for Analysis and Numerics

WWU Münster

Tel.: +49 251 83-35091

a.schlichting@uni-muenster.de

## News and Updates

- Preprint: Graph-to-local limit for the nonlocal interaction equation
Together with Antonio Esposito and Georg Heinze, we study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretisation for the equation under study.

See arXiv:2306.03457 - Preprint: Variational convergence of the Scharfetter-Gummel scheme
Together with Anastasiia Hraivoronska and Oliver Tse, we explore the convergence of the Scharfetter-Gummel scheme for the aggregation-diffusion equation using a variational approach. Our investigation involves obtaining a novel gradient structure for the finite volume scheme that works consistently for any nonnegative diffusion constant, which allows us to study the discrete-to-continuum and zero-diffusion limits simultaneously. The zero-diffusion limit for the Scharfetter-Gummel scheme corresponds to the upwind finite volume scheme for the aggregation equation. In both cases, we establish a convergence result in terms of gradient structures, recovering the Otto gradient flow structure for the aggregation-diffusion equation based on the 2-Wasserstein distance.

See arxiv.org/abs/2306.02226 - Published: On a class of nonlocal continuity equations on graphs:
The article with Antonio Esposito and Francesco Saverio Patacchini on

*a class of nonlocal continuity equations on graphs*got published in the European Journal of Applied Mathematics. This is a follow-up work on our previous work also with Dejan Slepcev, where we introduced evolutions on graphs based on Upwind interpolation. In this work, we look at more general interpolation functions and provide a well-posedness theory based on a fixed point argument. - Published: Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients
The paper with Víctor Navarro-Fernández on Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients got published at ESAIM: Mathematical Modelling and Numerical Analysis (M2AN). In the revision (also on arXiv:2201.10411), we arrived at uniform errors estimate in the diffusion constant also in the limit of vanishing diffusion.

- Preprint: Covariance-modulated optimal transport and gradient flowsTogether with Martin Burger, Franca Hoffmann, Daniel Matthes and Matthias Erbar, we investigate a new dynamical optimal transport distance in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that […]