A fast stochastic interacting particle-field method for 3D parabolic parabolic Chemotaxis systems: numerical algorithms and error analysis

In this paper, we develop a novel numerical framework, namely the stochastic interacting particle-field method with particle-in-cell acceleration (SIPF-PIC), for the efficient simulation of the three-dimensional (3D) parabolic-parabolic Keller-Segel (KS) systems. The SIPF-PIC method integrates Lagrangian particle dynamics with spectral field solvers by leveraging localized particle-grid interpolations and fast Fourier transform (FFT) techniques. For $P$ particles and $H$ Fourier modes per spatial dimension, the SIPF-PIC method achieves a computational complexity of $O(P + H^3 \log H)$ per time step, a significant improvement over the original SIPF method (proposed in \cite{SIPF1}), which has a computational complexity of $O(PH^3)$, while preserving numerical accuracy. Moreover, we carry out a rigorous error analysis for the proposed method and establish the corresponding error estimates. Finally, we present numerical experiments to validate the convergence order and demonstrate the computational efficiency of SIPF-PIC. Further numerical experiments show the method’s capability of capturing complex blowup dynamics beyond single-point collapse, including ring-shaped singularities.

💡 Research Summary

The paper introduces a new computational framework called SIPF‑PIC (Stochastic Interacting Particle‑Field with Particle‑in‑Cell acceleration) for the three‑dimensional parabolic‑parabolic Keller‑Segel (KS) system, a nonlinear PDE model describing chemotactic aggregation of cells or microorganisms. Traditional mesh‑based schemes struggle with the high resolution and long‑time integration required to capture finite‑time blow‑up phenomena, while the previously proposed stochastic interacting particle‑field (SIPF) method suffers from an O(P H³) per‑step cost because each of the P particles interacts with every one of the H³ Fourier modes of the chemoattractant field. This cost becomes prohibitive for realistic 3D simulations with millions of particles and fine spectral grids.

SIPF‑PIC overcomes this bottleneck by embedding a particle‑in‑cell (PIC) strategy into the SIPF algorithm. The method consists of two stages at each time step: (1) particle‑to‑grid projection, where particle positions are deposited onto a uniform periodic grid; this operation is mathematically equivalent to an unequally spaced fast Fourier transform (USFFT) and allows the global spectral coupling to be computed with a fast Fourier transform (FFT) in O(H³ log H) time. (2) grid‑to‑particle interpolation, where the spectral solution for the chemoattractant concentration is interpolated back to the particle locations using standard PIC interpolation kernels, incurring only O(P) cost. Consequently, the overall per‑step complexity reduces to O(P + H³ log H), enabling simulations with P > 10⁶ particles and H = 256 spectral modes per dimension.

The authors provide a rigorous error analysis. Under mild regularity assumptions on the exact solution (ρ, c) and the numerical chemoattractant field \hat{c}, Theorem 4.12 establishes a convergence bound in the Wasserstein‑1 distance: \