A Novel Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Chemotaxis System

Abstract: We introduce an efficient stochastic interacting particle-field (SIPF) algorithm with no history dependence for computing aggregation patterns and near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis system in three space dimensions (3D). The KS solutions are approximated as empirical measures of particles coupled with a smoother field (concentration of chemo-attractant) variable computed by the spectral method. Instead of using heat kernels causing history dependence and high memory cost, we leverage the implicit Euler discretization to derive a one-step recursion in time for stochastic particle positions and the field variable based on the explicit Green's function of an elliptic operator of the form Laplacian minus a positive constant. In numerical experiments, we observe that the resulting SIPF algorithm is convergent and self-adaptive to the high gradient part of solutions. Despite the lack of analytical knowledge (e.g. a self-similar ansatz) of the blowup, the SIPF algorithm provides a low-cost approach to study the emergence of finite time blowup in 3D by only dozens of Fourier modes and through varying the amount of initial mass and tracking the evolution of the field variable. Notably, the algorithm can handle at ease multi-modal initial data and the subsequent complex evolution involving the merging of particle clusters and formation of a finite time singularity.