Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit

Abstract

Abstract. In this paper we introduce numerical schemes for a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. In particular, we study the numerical passage of the Boltzmann equation with singular kernel to nonlinear friction equations in the so-called quasi elastic limit. To this aim we introduce a Fourier spectral method for the Boltzmann equation [25,26] and show that the kernel modes that define the spectral method have the correct quasi elastic limit providing a consistent spectral method for the limiting nonlinear friction equation.Mathematics Subject Classification. 65L60, 65R20, 76P05, 82C40.

Lagrangian Numerical Approximations to One‐Dimensional Convolution‐Diffusion Equations

“…The solution of (4.1) has been shown to converge weakly to the one of (2.1) as σ → 0 in [26] However, the results of [12] are more interesting at a computational level because they deal with smoother solutions. Let us recall:…”

“…First, it is customary now to decouple the transport and collision steps relying on a time-splitting technique, see e.g. [28,26,18]; thus it makes sense to restrict ourselves to homogeneous densities f (t, v). We shall therefore concentrate on the following Cauchy problem for t, v ∈ R + * × R, (1.1) where I stands for an even convex interaction potential.…”

“…This model was proposed by McNamara and Young [27] and then studied from a mathematical viewpoint in [1,2,7,13,14]. Roughly speaking, it describes a huge number of particles that undergo inelastic collisions, see [30,26], which lead to a monotonic decay of the temperature. Steady-states display a very elementary structure, namely a Dirac mass concentrated onto the null velocity; however, a rescaling process allows to derive a particular class of intermediate asymptotics called homogeneous cooling states, [2,24,30].…”

“…This trick is reminiscent of a previous work, [19], and allows to circumvent the issues which are connected to numerical approximation of probabillity measures with finitedifferences, see [9,17,26] for more in this direction. One feature of paramount importance for (1.5) is the easiness in proving contractivity in any p-Wasserstein metric for various numerical approximations, just relying on the standard Jensen's inequality.…”

A rescaling velocity method for dissipative kinetic equations. Applications to granular media

“…Another approach consists in a direct resolution of the Boltzmann operator on a phase space grid. For instance, deterministic and highly accurate methods based on a spectral discretization of the collisional operator have been proposed by F. Filbet, G. Naldi, L. Pareschi, G. Toscani and G. Russo in [39,37,24] for the space-homogeneous setting. Although being of complexity O(N 2 ), they are spectrally accurate and then need very few points to be precise.…”