Numerical schemes for kinetic equation with diffusion limit and anomalous time scale
Résumé
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 ε 2 (1 + |ln(ε)|). 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. The bifurcation between these two limits is the classical diffusion limit with anomalous time scale treated in this work. Our aim is to develop numerical schemes which work for the different regimes, with no restriction on the numerical parameters. Indeed, the degen-eracy ε → 0 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. Since the degeneracy towards the diffusion limit is very slow, it is also essential to respect the asymptotic behavior of the solution, and not only the limit. Various numerical tests are performed to illustrate the efficiency of our methods in this context.
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