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

Junge, Oliver; Matthes, Daniel; Osberger, Horst

doi:10.1137/16m1056560

Cited by 38 publications

(45 citation statements)

References 23 publications

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“…Since µ n 1 = µ n + c n 2 (µ n 1 − µ n 2 ), µ n 2 = µ n + c n 1 (µ n 2 − µ n 1 ), we deduce from (72) the desired L 2 ((0, T ); L q d (Ω)) estimates on the phase potentials µ n i with q d defined in Definition 2.3. Finally, the combination of the relations (36), (27), (53) and (41) yields…”

Section: Flow Interchange and Entropy Estimatementioning

confidence: 99%

A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow

Cancès

Matthes

Nabet

2019

Arch Rational Mech Anal

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We study a non-local version of the Cahn-Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. In difference to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes -but not necessarily the fluxes themselves -annihilate each other. Our main result is a rigorous proof of existence of weak solutions. The starting point is the formal representation of the dynamics as a constrained gradient flow in the Wasserstein metric. We then show that time-discrete approximations by means of the incremental minimizing movement scheme converge to a weak solution in the limit. Further, we compare the non-local model to the classical Cahn-Hilliard model in numerical experiments. Our results illustrate the significant speed-up in the decay of the free energy due to the higher degree of freedom for the velocity fields.∇ · J tot = 0 where J tot = J 1 + J 2 1991 Mathematics Subject Classification. 35K65, 35K41, 49J40, 76T99.

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Section: Flow Interchange and Entropy Estimatementioning

confidence: 99%

A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow

Cancès

Matthes

Nabet

2019

Arch Rational Mech Anal

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“…A common approach is to go to the Lagrangian formulation, using that the optimal γ is typically concentrated on the graph of a transport map T : Ω → Ω. This is extremely efficient in one space dimension [7,22,23], but becomes significantly more cumbersome -and difficult to analyze -in multiple dimensions [5,12,13,18]. Various alternatives to the Lagrangian approach are available, including finite volume methods [21], blob methods [11] etc.…”

Section: Discretization and Regularizationmentioning

confidence: 99%

Discretization of flux-limited gradient flows: $\Gamma $-convergence and numerical schemes

Matthes

Söllner

2019

Math. Comp.

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We study a discretization in space and time for a class of nonlinear diffusion equations with flux limitation. That class contains the so-called relativistic heat equation, as well as other gradient flows of Renyi entropies with respect to transportation metrics with finite maximal velocity. Discretization in time is performed with the JKO method, thus preserving the variational structure of the gradient flow. This is combined with an entropic regularization of the transport distance, which allows for an efficient numerical calculation of the JKO minimizers. Solutions to the fully discrete equations are entropy dissipating, mass conserving, and respect the finite speed of propagation of support.First, we give a proof of Γ-convergence of the infinite chain of JKO steps in the joint limit of infinitely refined spatial discretization and vanishing entropic regularization. The singularity of the cost function makes the construction of the recovery sequence significantly more difficult than in the L p -Wasserstein case. Second, we define a practical numerical method by combining the JKO time discretization with a "light speed" solver for the spatially discrete minimization problem using Dykstra's algorithm, and demonstrate its efficiency in a series of experiments.

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“…The approximate solution inherits automatically positivity, mass conservation and energy dissipation. Different approaches to actually compute the minimizers of (1.4) have been investigated: particle schemes [6,8,7,38]; evolving diffeomorphisms [8,10]; Lagrangian schemes [3,12,14,19,26,28]; entropic regularization [31]. However, it turns out that the application of these schemes to gradient flows in L 2 -Wasserstein space is intricate, since computing the L 2 -Wasserstein distance and its gradient is difficult in dimension two or more.…”

Section: Introductionmentioning

confidence: 99%

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

Plazotta¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We prove convergence of a variational formulation of the BDF2 method applied to the nonlinear Fokker-Planck equation. Our approach is inspired by the JKO-method and exploits the differential structure of the underlying L 2 -Wasserstein space. The technique presented here extends and strengthens the results of our own recent work on the BDF2 method for general metric gradient flows in the special case of the non-linear Fokker-Planck equation: firstly, we do not require uniform semi-convexity of the augmented energy functional; secondly, we prove strong instead of merely weak convergence of the timediscrete approximations; thirdly, we directly prove without using the abstract theory of curves of maximal slope that the obtained limit curve is a weak solution of the non-linear Fokker-Planck equation.

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

Cited by 38 publications

References 23 publications

A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow

A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow

Discretization of flux-limited gradient flows: $\Gamma $-convergence and numerical schemes

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

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