A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs

Abstract

In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for the kinetic chemotaxis system with random inputs, which will converge to the modified Keller-Segel model with random inputs in the diffusive regime. Based on the generalized Polynomial Chaos (gPC) approach, we design a high order stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time discretization with a macroscopic penalty term. The new schemes improve the parabolic CFL condition to a hyperbolic type when the mean free path is small, which shows significant efficiency especially in uncertainty quantification (UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property will be shown asymptotically and verified numerically in several tests. Many other numerical tests are conducted to explore the effect of the randomness in the kinetic system, in the aim of providing more intuitions for the theoretic study of the chemotaxis models.

Diffusive limit of a two-dimensional well-balanced approximation to a kinetic model of chemotaxis

“…Numerical diffusive limits toward Patlak-Keller-Segel parabolic systems were extensively studied following [6,17,35]: see for instance, [1,15,31]. Various blowup phenomena for the asymptotic diffusive system were displayed in [18].…”

A bi-fidelity stochastic collocation method for transport equations with diffusive scaling and multi-dimensional random inputs

“…Furthermore, an empirical error bound estimation is computed to access the quality of the bi-fidelity approximation. Further researches will consider the extension of the present approach to transport equations occurring biomathematics, like chemotaxis and epidemiology [3,4,7,25].…”

“…Besides the multiscale challenges, practical applications of the linear transport model usually contain uncertainties [4,25,26,30]. For example, the scattering cross-section in the collision operator is usually extracted from data, or in some cases we have only a rough estimate of the initial data.…”