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

Benjamin Söllner; Oliver Junge

doi:10.3934/cpaa.2020190

Article Contents

2020, Volume 19, Issue 9: 4227-4256. Doi: 10.3934/cpaa.2020190

This issue Previous Article Next Article On nonexistence of extremals for the Trudinger-Moser functionals involving

$ L^p $

norms

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

Benjamin Söllner^, and
Oliver Junge

Department of Mathematics, Technical University of Munich, 85747 Garching, Germany

^*Corresponding author
^*Corresponding author

Received: June 30, 2019

Revised: December 31, 2019

Published: June 2020

This research was supported by the German Research Foundation (DFG), Collaborative Research Center SFB-TR 109

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Abstract

We study a Lagrangian numerical scheme for solving a nonlinear drift diffusion equations of the form $ \partial_t u = \partial_x(u \cdot ({\sf c}^*)^\prime[\partial_x \mathit{h}^\prime(u)+ \mathit{v}^\prime]) $, like Fokker-Plank and $ q $-Laplace equations, on an interval. This scheme will consist of a spatio-temporal discretization founded on the formulation of the equation in terms of inverse distribution functions. It is based on the gradient flow structure of the equation with respect to optimal transport distances for a family of costs that are in some sense $ p $-Wasserstein like. Additionally we will show that, under a regularity assumption on the initial data, this also includes a family of discontinuous, flux-limiting cost inducing equations like Rosenau's relativistic heat equation. We show that this discretization inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, a minimum/maximum principle and flux-limitation in the case of the corresponding cost. Convergence in the limit of vanishing mesh size will be proven as the main result. Finally we will present numerical experiments including a numerical convergence analysis.

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Mathematics Subject Classification: Primary: 35K30, 35Q99, 65M12; Secondary: 35B40.

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Figure 1. Experiment: p-Wasserstein cost, linear diffusion. Left: Approximate densities $ u(t, \cdot) $ at $ t = 0.01\cdot 10^k $, $ k = 0, 0.12, 0.24, \ldots, \log_{10}(200) $, initial mass uniformly distributed on $ [-0.3, 0.3] $. Right: the corresponding characteristics

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Figure 2. Experiment: p-Wasserstein cost, linear diffusion. Left: Approximate densities $ u(t, \cdot) $ at $ t = 0.01\cdot 10^k $, $ k = 0, 0.12, 0.24, \ldots, \log_{10}(200) $, initial mass uniformly distributed on $ [-3, -2.4] $. Right: the corresponding characteristics

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Figure 3. Experiment: relativistic cost, linear diffusion. Left: Approximate densities $ u(t, \cdot) $ for $ t = 0.01\cdot 10^k $, $ k = 0, 0.12, 0.24, \ldots, \log_{10}(200) $, initial mass uniformly distributed on $ [-0.3, 0.3] $. Right: the corresponding characteristics (dashed: speed of light)

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Figure 4. Convergence analysis: relativistic cost, linear diffusion. $ L^1 $-error of the inverse distribution function in dependence of the grid size (left), and in dependence of the time step (right)

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Figure 5. Experiment: $ q $-Laplace ($ p = \frac43, m = \frac53 $). Left: Approximate densities $ u(t,\cdot) $ for $ t = 0.01\cdot 10^k $, $ k = 0,0.12,0.24,\ldots,\log_{10}(200) $, initial mass uniformly distributed on $ [-0.3,0.3] $. Right: the corresponding characteristics(dashed: speed of light)

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