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

TL;DR

This paper introduces a novel numerical method for efficiently computing drift and diffusion coefficients in the diffusion approximation of kinetic equations, using eigenvalues and eigenfunctions of a Schrödinger operator.

Contribution

The paper presents a new approach based on spectral analysis of a Schrödinger operator to calculate coefficients in kinetic equations more efficiently.

Findings

Method accurately computes coefficients in fewer iterations.

Numerical simulations confirm the efficiency and accuracy of the approach.

Approach outperforms traditional methods in computational speed.

Abstract

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 Schr\"odinger operator. These theoretical results are supported by numerical simulations showcasing the efficiency of the method.