We aim to efficiently compute spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion, and mutation/selection through a fitness function originated in the FK semigroup. We analyze convergence of the algorithm based on operator splitting, present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed.

翻译：本文旨在高效计算无散随机流中反应-扩散-平流（RDA）波前在Kolmogorov-Petrovsky-Piskunov（KPP）非线性条件下的传播速度。我们研究了一种随机交互粒子法（IPM），用于求解具有空间随机系数的线性平流-扩散算子所对应的约化主特征值（李雅普诺夫指数）问题。该方法以随机平流场的傅里叶表示和主特征值（李雅普诺夫指数）的Feynman-Kac（FK）公式为基础，通过遗传进化算法实现。粒子经历平流-扩散过程，并通过源自FK半群的适应度函数进行突变/选择。我们基于算子分裂分析了算法的收敛性，并给出了典型流动（如二维胞流和三维Arnold-Beltrami-Childress（ABC）流在随机扰动下）的数值结果。二维算例与半拉格朗日计算进行了一致性验证。三维结果表明：IPM方法具有无网格和自适应的特性，在无界区域的高维随机流中，对于以平流为主导的体系，无需截断处理即可简单高效地计算波前传播速度。