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

TL;DR

This paper introduces a high-order stochastic asymptotic-preserving scheme for kinetic chemotaxis models with random inputs, improving efficiency and accuracy in multiscale uncertainty quantification tasks.

Contribution

It develops a novel stochastic Galerkin method with IMEX Runge-Kutta discretization that maintains asymptotic-preserving properties in chemotaxis models with randomness.

Findings

The scheme converges to the modified Keller-Segel model with random inputs.

It improves the CFL condition from parabolic to hyperbolic type.

Numerical tests validate the scheme's efficiency and accuracy in uncertainty quantification.

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.