We propose a stochastic branching particle-based method for solving nonlinear non-conservative advection-diffusion-reaction equations. The method splits the evolution into an advection-diffusion step, based on a linearized Kolmogorov forward equation and approximated by stochastic particle transport, and a reaction step implemented through a branching birth-death process that provides a consistent temporal discretization of the underlying reaction dynamics. This construction yields a mesh-free, nonnegativity-preserving scheme that naturally accommodates non-conservative systems and remains robust in the presence of singularities or blow-up. We validate the method on two representative two-dimensional systems: the Allen-Cahn equation and the Keller-Segel chemotaxis model. In both cases, the present method accurately captures nonlinear behaviors such as phase separation and aggregation, and achieves reliable performance without the need for adaptive mesh refinement.

翻译：我们提出了一种基于随机分支粒子的方法，用于求解非线性非守恒对流-扩散-反应方程。该方法将演化过程分解为对流-扩散步骤和反应步骤：对流-扩散步骤基于线性化的Kolmogorov前向方程，通过随机粒子输运进行近似；反应步骤则通过分支生灭过程实现，为底层反应动力学提供了一致的时间离散化。这一构造产生了一种无网格、保持非负性的格式，能自然地适应非守恒系统，并在存在奇点或爆破现象时保持鲁棒性。我们在两个典型的二维系统上验证了该方法：Allen-Cahn方程和Keller-Segel趋化模型。在两种情况下，本方法均能准确捕捉相分离和聚集等非线性行为，且无需自适应网格细化即可实现可靠的性能。