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

TL;DR

This work introduces a memory-efficient SIPF algorithm for the fully parabolic KS system in 3D by coupling an empirical-particle representation of the density with a spectrally discretized chemoattractant field. An implicit-Euler update for the field via the Green's function of the operator $\nabla^2 - k^2$ enables a one-step, history-free recursion, while the particle dynamics are advanced with Euler–Maruyama and velocity computed from the convolved field. The method demonstrates first-order temporal convergence, captures near-singular aggregation, and reveals how mass, geometry, and parameters influence blow-up, including non-radial and multi-cluster initial data scenarios. Although memory efficiency and spectral accuracy are strengths, the approach incurs higher costs when many Fourier modes are needed to resolve sharp blow-up features, pointing to future work on adaptivity and acceleration. Overall, SIPF provides a practical, scalable framework to study finite-time blow-up and aggregation in 3D KS systems where traditional grid-based methods are prohibitively expensive.

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.

Figures (13)

• Figure 1: Density $\rho$ approximated by empirical distribution at $T=0.1$: the mass effect on focusing.
• Figure 2: Chemical concentration $c$ at final time $T=0.1$, sliced at $z=0$: the mass effect on focusing.
• Figure 3: Maximum of chemical concentration $c$ vs computation time $T$ with different total mass $M_0$: identifying blow up by refining the discretization.
• Figure 4: Maximum of $c$ scales linearly with the number of Fourier modes $H$ (in each dimension) under total mass $M_0=80$ (super critical): a $\delta$ type blow up.
• Figure 5: Relative errors of $c$ vs. $\delta t$, compared with $\delta= 2^{-11}\times 0.01$. Fitted rate: $e(\delta t)=\mathcal{O}(\delta t^{1.011})$.

Theorems & Definitions (4)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 4.1