沉降过程的高效数值算法研究

沉降过程通常由群体平衡方程来描述，它包括流体的方程、化学反应方程及粒子尺寸分布方程。各个方程之间的高度耦合、时间相关对流占优对流-扩散-反应方程的求解及粒子尺寸分布方程的高维特点等困难使得设计相关数值算法面临巨大挑战。本项目拟研究其中的核心问题，以期在沉降过程的数值模拟方面取得突破。具体来说，本项目将研究各向异性网格条件下时间相关对流占优对流-扩散-反应方程的稳定化有限元方法；对粒子尺寸分布方程设计基于算子分裂方法和特征线方法的并行算法；结合随机过程理论，对粒子破裂和聚并过程做更精细的数学分析并将其耦合于粒子尺寸分布方程的数值求解过程中；最终对沉降过程设计完整的数值算法。

Precipitation processes are modeled by population balance systems consisting of equations describing the flow field, equations for the chemical reaction and an equation for the particle size distribution. The main difficulties for simulating the precipitation processes include the coupling of these equations, the solving of the time dependent convection-dominated convection-diffusion-reaction equations and the high dimensional property of the particle size distribution equation, which make the numerical simulation for the precipitation processes a challenging task. In this project, we will do some research on the core problems mentioned above. To be specific, we will study the stabilized finite element methods for the time dependent convection-dominated convection-diffusion-reaction equations on anisotropic meshes, design the parallel algorithms based on operator-splitting method and the method of characteristics for the particle size distribution equation, make finer mathematical analysis for the aggregation and breakage of particles via the theory of stochastic process and then insert it into the numerical solving process for the particle size distribution equation, and finally design the full numerical algorithm for the precipitation processes.