# B09Structure Preserving Discretization of Gradient Flows

## Analyzing Discrete Curves of Steepest Descent in Discrete Energy Landscapes

Many evolution equations from physics, like those for diffusion of dissolved substances or for phase separation in alloys, describe processes in which a system tries to reach the minimum of its energy (or the like) as quickly as possible. The descent towards the minimum is restrained by the system’s inertia. In this project, we discretize these evolution equations by optimizing discrete curves in a discrete energy landscape with respect to a discrete inertia tensor.

#### Scientific Details+

The project's objective is to investigate - analytically and numerically - the properties of adapted full discretizations for a class of evolution equations with gradient flow structure. Our motivation is two-fold. One goal is to define novel schemes for numerical solution of the evolution equations, that are stable and convergent, and in addition preserve various aspects of the variational structure and the associated qualitative properties of solutions on the discrete level. The second goal is to perform a qualitative analysis, e.g., to study the long-time behaviour, of the resulting dynamical system on a discrete space.

We are mainly interested in the discretization of fluid type equations for mass densities which arise as a steepest descent of an energy or entropy functional in the Wasserstein distance or a related mass transport metric. Specific examples that we intend to study are second order porous medium and fourth order lubrication equations, as well as, in later phases of the project, also more general equations of Cahn-Hilliard type and also genuine Euler-like hydrodynamic models. Some of the expected advantages of our structure preserving approach in comparison to a "generic" discretization are the following: the potential of the flow will be time-monotone; contraction estimates in the underlying metric will be preserved; and auxiliary Lyapunov functionals for the continuous flow will have discrete counterparts. These properties will also pave the way to proving stability and convergence. On top of that, the discretization will automatically guarantee such general features as non-negativity of solutions and the conservation of mass.

We plan to explore two different strategies to define adapted discretizations, which we label Lagrangian and Eulerian. The first uses a formulation of the underlying fluid dynamics in a co-moving frame. That is, we discretize the evolution of Lagrangian maps rather than the temporal change of the density directly. This approch is well-suited to gradient flows in the Wasserstein metric, where the Lagrangian map is a concatenation of infinitesimal plans for optimal mass transfer. Our discretization preserves this intuitive geometric interpretation. In the Eulerian approach, which is close to a classical finite element discretization, we will approximate the continuous densities by functions in a finite-dimensional ansatz space, e.g., by piecewise constant functions with respect to a fixed decomposition of the spatial domain. For a truely variational approach on these grounds, we will need to design of an efficient method for the calculation of the transportation distance between functions in the ansatz space.

The dynamics of the resulting discrete dynamical system will then be studied - analytically and numerically - in view of the following:
• energy dissipation, Lyapunov functionals, and derivation of discrete a priori estimates;
• qualitative properties of solutions like strict positivity and growth of the support;
• rate of equilibration in the long-time limit;
• existence and stability of special solutions, like quasi-self-similar profiles;
• convergence of the discrete solutions to a continuous one.

#### Publications+

##### Papers
• Simon Plazotta.
A BDF2-approach for the non-linear Fokker-Planck equation.
Discrete Contin. Dyn. Syst., 39(5):2893–2913, May 2019.
arXiv:1801.09603, doi:10.3934/dcds.2019120.
• Oliver Junge and Benjamin Söllner.
A convergent Lagrangian discretization for $p$-Laplace and flux-limited diffusion equations.
preprint, 2019.
arXiv:1906.01321.
• Flore Nabet Clément Cancès, Daniel Matthes.
A two-phase two-fluxes degenerate Cahn-Hilliard model as constrained Wasserstein gradient flow.
Arch. Ration. Mech. Anal., 233(2):837–866, 2019.
arXiv:1712.06446, doi:10.1007/s00205-019-01369-6.
• Daniel Matthes and Benjamin Söllner.
Discretization of Flux-Limited Gradient Flows: Γ-convergence and numerical schemes.
preprint, 2019.
arXiv:1910.09843.
• Julian Fischer and Daniel Matthes.
The waiting time phenomenon in spatially discretized porous medium and thin film equations.
preprint at arXiv, 2019.
arXiv:1911.04185.
• Daniel Matthes and Simon Plazotta.
A Variational Formulation of the BDF2 Method for Metric Gradient Flows.
ESAIM: M2AN, Forthcoming article, July 2018.
doi:10.1051/m2an/2018045.
• Daniel Matthes José A. Carrillo, Bertram Düring and David S. McCormick.
A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes.
Journal of Scientific Computing, June 2018, Volume 75, Issue 3, pp 1463–1499, November 2017.
doi:10.1007/s10915-017-0594-5.
• Oliver Junge and Ioannis G Kevrekidis.
On the sighting of unicorns: A variational approach to computing invariant sets in dynamical systems.
Chaos: An Interdisciplinary Journal of Nonlinear Science, Nr. 6, Volume 27, June 2017.
doi:10.1063/1.4983468.
• Daniel Matthes and Benjamin Söllner.
Convergent Lagrangian Discretization for Drift-Diffusion with Nonlocal Aggregation.
Innovative Algorithms and Analysis pp 313-351, March 2017.
doi:10.1007/978-3-319-49262-9_12.
• Daniel Matthes Oliver Junge and Horst Osberger.
A Fully Discrete Variational Scheme for Solving Nonlinear Fokker-Planck Equations in Multiple Space Dimensions.
SIAM Journal on Numerical Analysis, 55(1):419–443, 2017.
arXiv:1509.07721, doi:10.1137/16M1056560.
• Simon Plazotta and Jonathan Zinsl.
High-frequency limit of non-autonomous gradient flows of functionals with time-periodic forcing.
Journal of Differential Equations, Vol. 261, Issue 12, 15 December 2016, Pages 6806-6855, December 2016.
doi:10.1016/j.jde.2016.09.003.
• Jan Maas and Daniel Matthes.
Long-Time Behavior of a Finite Volume Discretization for a Fourth Order Diffusion Equation.
Nonlinearity, Volume 29, Number 7, June 2016.
URL: http://iopscience.iop.org/article/10.1088/0951-7715/29/7/1992/meta.
• Oliver Junge Andres Denner and Daniel Matthes.
Computing coherent sets using the Fokker-Planck equation.
Journal of Computational Dynamics, 2016, Vol. 3, Issue 2, 2016.
doi:10.3934/jcd.2016008.
• J-F Mennemann, D Matthes, R-M Weishäupl, and T Langen.
Optimal control of Bose-Einstein condensates in three dimensions.
New Journal of Physics, 17(11):113027, November 2015.
URL: http://stacks.iop.org/1367-2630/17/i=11/a=113027.
• Daniel Matthes and Horst Osberger.
A convergent Lagrangian discretization for a nonlinear fourth order equation.
Found. Comput. Math., 2015. online first.
arXiv:1410.1728.
• Horst Osberger and Daniel Matthes.
Convergence of a Fully Discrete Variational Scheme for a Thin Film Equation.
Radon Series on Computational and Applied Mathematics, 2015. accepted.
arXiv:1509.01513.
• Jonathan Zinsl and Daniel Matthes.
Exponential Convergence to Equilibrium in a Coupled Gradient Flow System Modelling Chemotaxis.
Analysis & PDE, 8(2):425–466, 2015.
arXiv:1310.3977.
• Horst Osberger.
Long-Time Behaviour of a Fully Discrete Lagrangian Scheme for a Family of Fourth Order.
Submitted, 2015.
arXiv:1501.04800.
• Jonathan Zinsl and Daniel Matthes.
Transport Distances and Geodesic Convexity for Systems of Degenerate Diffusion Equations.
Calculus of Variations and Partial Differential Equations, 2015. accepted.
arXiv:1409.6520.
• Marco Di Francesco, Massimo Fornasier, Jan-Christian Hütter, and Daniel Matthes.
Asymptotic Behavior of Gradient Flows Driven by Nonlocal Power Repulsion and Attraction Potentials in One Dimension.
accepted at SIAM-MA, January 2014.
arXiv:1401.2338.
• Daniel Matthes and Horst Osberger.
Convergence of a Variational Lagrangian Scheme for a Nonlinear Drift Diffusion Equation.
ESAIM: Mathematical Modelling and Numerical Analysis, 48(03):697–726, 2014. Cambridge Univ Press.
arXiv:1301.0747.
• Jonathan Zinsl.
Geodesically Convex Energies and Confinement of Solutions for a Multi-Component System of Nonlocal Interaction Equations.
Submitted, 2014.
arXiv:1412.3266.

##### PhD thesis
• H. Osberger.
Fully variational Lagrangian discretizations for second and fourth order evolution equations.
Dissertation, Technische Universität München, September 2015.
URL: https://mediatum.ub.tum.de/1271651, dgd:182.

#### Prof. Dr. Daniel Matthes   +

Projects: B09
University: TU München
E-Mail: matthes[at]ma.tum.de
Website: http://www-m8.ma.tum.de/personen/matthes/

#### Prof. Dr. Oliver Junge   +

Projects: B09, B12
University: TU München
E-Mail: oj[at]tum.de
Website: http://www-m3.ma.tum.de/Allgemeines/OliverJunge

#### Eva-Maria Rott   +

Projects: B09
University: TU München, M8, 03.06.039
Address: Boltzmannstraße 3, 85748 Garching, GERMANY
Tel: +49 89 28918336
E-Mail: eva-maria.rott[at]tum.de
Website: https://www-m8.ma.tum.de/bin/view/Allgemeines/Eva-MariaRott