Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations

V. Bonnaillie-Noël, J. A. Carrillo, T. Goudon, G. A. Pavliotis
2016 IMA Journal of Numerical Analysis  
In this paper we study the diffusion approximation of a swarming model given by a system of interacting Langevin equations with nonlinear friction. The diffusion approximation requires the calculation of the drift and diffusion coefficients that are given as averages of solutions to appropriate Poisson equations. We present a new numerical method for computing these coefficients that is based on the calculation of the eigenvalues and eigenfunctions of a Schrödinger operator. These theoretical
more » ... sults are supported by numerical simulations showcasing the efficiency of the method. Here, f ε (t, r, v) denotes the distribution in phase space of a certain population (of particles, individuals, . . .); r ∈ R d , v ∈ R d stand for the space and velocity variables, respectively. Equation (1.1) is written in dimensionless form and we consider the regime where the parameter ε > 0, which depends on the typical length and time scales of the phenomena under consideration, is small. The potential (t, r) → Φ(t, r) can be defined self-consistently, typically by a convolution with the macroscopic density f ε dv. On the Owing to the linearity of Q, we rewrite this as a convection-diffusion equation: where the coefficients are defined by the matrices with Q −1 the pseudo-inverse of Q, which has to be properly defined on the orthogonal of Span{F}. In particular, the dissipative nature of Q implies that D is non-negative. There is a huge literature on the analysis of such asymptotic problems, motivated by various application fields (radiative transfer theory, neutron transport, the modelling of semiconductors, population dynamics, etc.); we refer the reader for instance to Bouchut et al.
doi:10.1093/imanum/drv066 fatcat:yxjpsmr3gjcgzdoxpgsvoaobcq