A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation

2016

Nonlinearity

Self Cite

Abstract. We consider a non-standard finite-volume discretization of a strongly non-linear fourth order diffusion equation on the d-dimensional cube, for arbitrary d ≥ 1. The scheme preserves two important structural properties of the equation: the first is the interpretation as a gradient flow in a mass transportation metric, and the second is an intimate relation to a linear Fokker-Planck equation. Thanks to these structural properties, the scheme possesses two discrete Lyapunov functionals. These functionals approximate the entropy and the Fisher information, respectively, and their dissipation rates converge to the optimal ones in the discrete-to-continuous limit. Using the dissipation, we derive estimates on the long-time asymptotics of the discrete solutions. Finally, we present results from numerical experiments which indicate that our discretization is able to capture significant features of the complex original dynamics, even with a rather coarse spatial resolution.

mentioning

confidence: 99%

“…However, here we do not use the Lagrangian structure behind (1) -which was essential in [34,39] -but define a scheme on grounds of a finite-volume discretization. Our ansatz is motivated by a particular structure-preserving discretization of linear Fokker-Planck equations, which has been introduced simultaneously in [8,31,37].…”

mentioning

confidence: 99%

Long-time behavior of a finite volume discretization for a fourth order diffusion equation

Maas

2016

Nonlinearity

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“…Since our model has a Wasserstein gradient flow structure, it would be natural to use a Lagrangian method as for instance in [7,31,27,15]. The main problem with this approach is that both phase move with their own speed, therefore such an approach would impose to move two meshes simultaneously.…”

Section: 3mentioning

confidence: 99%

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

Cancès

Arch Rational Mech Anal

Nabet

2019

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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.

“…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

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.