# 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
###### A BDF2-approach for the non-linear Fokker-Planck equation

Author: Plazotta, Simon
Journal: Discrete Contin. Dyn. Syst., 39(5):2893--2913
Date: May 2019
DOI: 10.3934/dcds.2019120

###### A convergent Lagrangian discretization for $p$-Laplace and flux-limited diffusion equations

Authors: Junge, Oliver and Söllner, Benjamin
Note: preprint
Date: 2019

###### A two-phase two-fluxes degenerate Cahn-Hilliard model as constrained Wasserstein gradient flow

Author: Clément Cancès, Daniel Matthes, Flore Nabet
Journal: Arch. Ration. Mech. Anal., 233(2):837--866
Date: 2019
DOI: 10.1007/s00205-019-01369-6

###### Discretization of Flux-Limited Gradient Flows: $\Gamma$-convergence and numerical schemes

Authors: Matthes, Daniel and Söllner, Benjamin
Note: preprint
Date: 2019

###### The waiting time phenomenon in spatially discretized porous medium and thin film equations

Authors: Fischer, Julian and Matthes, Daniel
Note: preprint at arXiv
Date: 2019

###### A Variational Formulation of the BDF2 Method for Metric Gradient Flows

Authors: Daniel Matthes and Plazotta, Simon
Journal: ESAIM: M2AN, Forthcoming article
Date: Jul 2018
DOI: 10.1051/m2an/2018045

###### A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes

Authors: José A. Carrillo, Bertram Düring, Daniel Matthes and McCormick, David S.
Journal: Journal of Scientific Computing, June 2018, Volume 75, Issue 3, pp 1463–1499
Date: Nov 2017
DOI: 10.1007/s10915-017-0594-5

###### On the sighting of unicorns: A variational approach to computing invariant sets in dynamical systems

Authors: Junge, Oliver and Kevrekidis, Ioannis G
Journal: Chaos: An Interdisciplinary Journal of Nonlinear Science, Nr. 6, Volume 27
Date: Jun 2017
DOI: 10.1063/1.4983468

###### Convergent Lagrangian Discretization for Drift-Diffusion with Nonlocal Aggregation

Authors: Matthes, Daniel and Söllner, Benjamin
Journal: Innovative Algorithms and Analysis pp 313-351
Date: Mar 2017
DOI: 10.1007/978-3-319-49262-9_12

###### A Fully Discrete Variational Scheme for Solving Nonlinear Fokker-Planck Equations in Multiple Space Dimensions

Authors: Oliver Junge, Daniel Matthes and Osberger, Horst
Journal: SIAM Journal on Numerical Analysis, 55(1):419-443
Date: 2017
DOI: 10.1137/16M1056560

###### High-frequency limit of non-autonomous gradient flows of functionals with time-periodic forcing

Authors: Plazotta, Simon and Zinsl, Jonathan
Journal: Journal of Differential Equations, Vol. 261, Issue 12, 15 December 2016, Pages 6806-6855
Date: Dec 2016
DOI: 10.1016/j.jde.2016.09.003

###### Long-Time Behavior of a Finite Volume Discretization for a Fourth Order Diffusion Equation

Authors: Maas, Jan and Matthes, Daniel
Journal: Nonlinearity, Volume 29, Number 7
Date: Jun 2016

###### Computing coherent sets using the Fokker-Planck equation

Authors: Andres Denner, Oliver Junge and Matthes, Daniel
Journal: Journal of Computational Dynamics, 2016, Vol. 3, Issue 2
Date: 2016
DOI: 10.3934/jcd.2016008

###### Optimal control of Bose-Einstein condensates in three dimensions

Authors: Mennemann, J-F and Matthes, D and Weishäupl, R-M and Langen, T
Journal: New Journal of Physics, 17(11):113027
Date: Nov 2015

###### A convergent Lagrangian discretization for a nonlinear fourth order equation

Authors: Matthes, Daniel and Osberger, Horst
Journal: Found. Comput. Math.
Note: online first
Date: 2015

###### Convergence of a Fully Discrete Variational Scheme for a Thin Film Equation

Authors: Osberger, Horst and Matthes, Daniel
Journal: Radon Series on Computational and Applied Mathematics
Note: accepted
Date: 2015

###### Exponential Convergence to Equilibrium in a Coupled Gradient Flow System Modelling Chemotaxis

Authors: Zinsl, Jonathan and Matthes, Daniel
Journal: Analysis & PDE, 8(2):425-466
Date: 2015

###### Long-Time Behaviour of a Fully Discrete Lagrangian Scheme for a Family of Fourth Order

Author: Osberger, Horst
Note: Submitted
Date: 2015

###### Transport Distances and Geodesic Convexity for Systems of Degenerate Diffusion Equations

Authors: Zinsl, Jonathan and Matthes, Daniel
Journal: Calculus of Variations and Partial Differential Equations
Note: accepted
Date: 2015

###### Asymptotic Behavior of Gradient Flows Driven by Nonlocal Power Repulsion and Attraction Potentials in One Dimension

Authors: Francesco, Marco Di and Fornasier, Massimo and Hütter, Jan-Christian and Matthes, Daniel
Note: accepted at SIAM-MA
Date: Jan 2014

###### Convergence of a Variational Lagrangian Scheme for a Nonlinear Drift Diffusion Equation

Authors: Matthes, Daniel and Osberger, Horst
Journal: ESAIM: Mathematical Modelling and Numerical Analysis, 48(03):697-726
Note: Cambridge Univ Press
Date: 2014

###### Geodesically Convex Energies and Confinement of Solutions for a Multi-Component System of Nonlocal Interaction Equations

Author: Zinsl, Jonathan
Note: Submitted
Date: 2014

##### PhD thesis
###### Fully variational Lagrangian discretizations for second and fourth order evolution equations

Author: Osberger, H.
Date: Sep 2015

#### Prof. Dr. Oliver Junge   +

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

#### 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/

#### Dr. Daniel Karrasch   +

Projects: Z, B09
University: TU München
E-Mail: daniel.karrasch[at]tum.de

#### Benjamin Söllner   +

Projects: B09
University: TU München
E-Mail: b.soellner[at]ma.tum.de