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

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    *Corresponding author 

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

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  • 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.

    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

    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

    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)

    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)

    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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