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.

翻译：我们提出了一种高效的、无历史依赖性的随机相互作用粒子-场（SIPF）算法，用于计算三维空间（3D）中抛物-抛物Keller-Segel（KS）趋化系统的聚集模式及近奇异解。KS解通过粒子的经验测度与由谱方法计算的平滑场变量（化学吸引物浓度）耦合来进行近似。为避免使用热核导致的历史依赖性和高内存成本，我们利用隐式欧拉离散化，基于椭圆算子（形式为拉普拉斯算子减去一个正常数）的显式格林函数，推导出随机粒子位置和场变量的单步时间递归。在数值实验中，我们观察到所得的SIPF算法是收敛的，并能自适应于解的高梯度区域。尽管缺乏对爆破的分析性知识（例如自相似假设），SIPF算法仅需数十个傅里叶模式，并通过改变初始质量大小以及跟踪场变量的演化，即可低成本地研究三维有限时间爆破的出现。值得注意的是，该算法能轻松处理多模态初始数据及其后续涉及粒子团簇合并和形成有限时间奇异性的复杂演化过程。