Numerical schemes for kinetic equation with diffusion limit and anomalous time scale

Hélène Hivert

doi:10.3934/krm.2018019

Article Contents

2018, Volume 11, Issue 2: 409-439. Doi: 10.3934/krm.2018019

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Numerical schemes for kinetic equation with diffusion limit and anomalous time scale

Hélène Hivert^,

ENS Lyon, UMPA UMR 5669 CNRS, and Inria Rhône-Alpes, projet NUMED, 46, allée d'Italie, 69364 Lyon Cedex 07, France

* Corresponding author
* Corresponding author

Received: October 2016

Revised: May 2017

Published: April 2018

The author is supported by ERC starting grant MESOPROBIO

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Abstract

In this work, we propose numerical schemes for linear kinetic equation, which are able to deal with a diffusion limit and an anomalous time scale of the form ${\varepsilon ^2}\left( {1 + \left| {\ln \left( \varepsilon \right)} \right|} \right)$. When the equilibrium distribution function is a heavy-tailed function, it is known that for an appropriate time scale, the mean-free-path limit leads either to diffusion or fractional diffusion equation, depending on the tail of the equilibrium. This work deals with a critical exponent between these two cases, for which an anomalous time scale must be used to find a standard diffusion limit. Our aim is to develop numerical schemes which work for the different regimes, with no restriction on the numerical parameters. Indeed, the degeneracy $ \varepsilon\to0$ makes the kinetic equation stiff. From a numerical point of view, it is necessary to construct schemes able to undertake this stiffness to avoid the increase of computational cost. In this case, it is crucial to capture numerically the effects of the large velocities of the heavy-tailed equilibrium. Moreover, we prove that the convergence towards the diffusion limit happens with two scales, the second being very slow. The schemes we propose are designed to respect this asymptotic behavior. Various numerical tests are performed to illustrate the efficiency of our methods in this context.

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Mathematics Subject Classification: Primary: 35B40, 65L04, 41A60; Secondary: 65M22.

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Figure 1. For $\Delta t = 10^{-2}$, the solutions of (34) at time $T = 0.1$ for different values of $\varepsilon$, when $a_\varepsilon$ is computed with (35), and the solution of the limit scheme (37)

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Figure 2. For $\Delta t = 10^{-2}$, the solutions of (34) at time $T = 0.1$ for different values of $\varepsilon$, when $a_\varepsilon$ is computed with (36), and the solution of the limit scheme (37)

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Figure 3. For $\Delta t = 10^{-2}$, the relative error between the solution of the scheme (34) and the limit scheme (37) at time $T = 0.1$, in function of $\varepsilon$ (log scale)

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Figure 6. For $\Delta t = 10^{-2}$, the solutions of IS, quasi-diff and diff schemes for different values of $\varepsilon$

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Figure 7. For $\Delta t = 10^{-2}$, the error (71) between the solution of IS and quasi-diff scheme in function of $\varepsilon$ (log scale)

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Figure 8. For $\Delta t = 10^{-4}$ and $\varepsilon = 1$, the solutions of the MMS scheme and of the explicit scheme (69)

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Figure 9. The relative consistency error (70) for the MMS scheme (log scale)

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Figure 10. For $\Delta t = 10^{-4}$, the solutions of MMS, quasi-diff and diff schemes for different values of $\varepsilon$

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Figure 4. For $\Delta t = 10^{-2}$ and $\varepsilon = 1$, the solutions of the IS scheme and of the explicit scheme (69)

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Figure 5. The relative consistency error (70) for the IS scheme (log scale)

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Figure 11. For $\Delta t = 10^{-4}$, the error (71) between the solution of MMS and quasi-diff scheme in function of $\varepsilon$ (log scale)

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Figure 12. For $\Delta t = 10^{-2}$ and $\varepsilon = 1$, the solutions of the DS scheme and of the explicit scheme (69)

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Figure 13. The relative consistency error (70) for the DS scheme (log scale)

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Figure 14. For $\Delta t = 10^{-2}$, the solutions of DS, quasi-diff and diff schemes for different values of $\varepsilon$

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Figure 15. For $\Delta t = 10^{-2}$, the error (71) between the solution of DS and quasi-diff scheme in function of $\varepsilon$ (log scale)

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Figure 16. The error (70) as a function of $\varepsilon$. The density $\rho_{reference}$ is the density given by the DS scheme for $\Delta t_{ref} = 5\cdot 10^{-5}$, and $\rho_{\Delta t}$ are the densities given by the DS scheme for different values of $\Delta t$ (log scale)

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