求解多维粒数衡算方程的高阶紧致差分方法  

High-order compact difference method for solving the multidimensional population balance equation

作　　者：李川 洪振取 单宝明[1] 徐啟蕾[1] 张方坤 LI Chuan;HONG Zhenqu;SHAN Baoming;XU Qiei;ZHANG Fangkun(College of Automation and Electronic Engineering,Qingdao University of Science&Technology,Qingdao 266061,Shandong,China)

机构地区：[1]青岛科技大学自动化与电子工程学院,山东青岛266061

出　　处：《化工学报》2024年第12期4513-4522,共10页CIESC Journal

基　　金：国家自然科学基金项目(62103216);山东省自然科学基金项目(ZR2020QF060)。

摘　　要：多维粒数衡算方程(MPBE)描述颗粒过程中两个或两个以上内在变量的粒度分布情况。由于大多数PBE缺乏解析解,通常需要采用计算昂贵的高阶或高分辨率(HR)方法来获得精确的数值解。如何高效、准确地获取数值解是其面临的挑战。针对以上问题,提出一种结合维度分裂的改进的高阶紧致差分(HOCD)方法,该方法能够使数值解具有四阶精度。采用维度分裂方法将多维问题分裂为几个一维问题,将分裂后的一维方程在时间和空间上进行离散,得到HOCD的三对角格式,并通过Tomas算法进行求解。在某些情况下,还进行了变量代换。除此之外,利用von Neumann稳定性分析方法证明了其稳定性。与HR方法相比,HOCD具有更高的计算精度和效率,且无严重的数值扩散。多个数值模拟证明了该方法的有效性。Multidimensional population balance equations(MPBE)describe the size distribution of a granular process over two or more intrinsic variables.Since most PBEs lack analytical solutions,computationally expensive high-order or high-resolution(HR)methods are often used to obtain accurate numerical solutions.How to obtain numerical solutions efficiently and accurately is a challenge it faces.To address the above problems,an improved higher-order compact difference(HOCD)method combining dimensional splitting is proposed,which enables the numerical solution with fourth-order accuracy.Dimensional splitting methods are used to split the multidimensional problem into several one-dimensional problems.The split one-dimensional equations are then discretized in space and time to produce the tridiagonal format of the HOCD,which may be solved by using the Tomas algorithm.Variable substitution is also carried out in some cases.Furthermore,the stability was demonstrated by using the von Neumann stability analysis method.Compared with HR methods,the HOCD has higher computational accuracy and computational efficiency without numerical diffusion.The effectiveness of this method is demonstrated by multiple numerical simulations.

关 键 词：粒数衡算方程 高阶紧致差分 粒度分布 维度分裂 稳定性分析 数值模拟 

分 类 号：TP391.9[自动化与计算机技术—计算机应用技术]