求解多维粒数衡算方程的高阶紧致差分方法

doi:10.11949/0438-1157.20240527

摘要/Abstract

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。

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

Abstract:

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.

Key words: population balance equation, high-order compact difference, size distribution, dimensional splitting, stability analysis, numerical simulation

中图分类号:

TP 391.9

李川, 洪振取, 单宝明, 徐啟蕾, 张方坤. 求解多维粒数衡算方程的高阶紧致差分方法[J]. 化工学报, 2024, 75(12): 4513-4522.

Chuan LI, Zhenqu HONG, Baoming SHAN, Qilei XU, Fangkun ZHANG. High-order compact difference method for solving the multidimensional population balance equation[J]. CIESC Journal, 2024, 75(12): 4513-4522.

图/表 6

表1 案例1不同网格尺寸的最大绝对误差及收敛阶

Table 1 Maximum absolute error and convergence rate with different grid sizes for case 1

h	HR		HOCD
h	MAE	Rate	MAE	Rate
0.04	25.408	—	4.395	—
0.02	11.809	1.105	0.282	3.964
0.01	4.246	1.476	1.42×10^-2	4.307
0.005	0.844	2.330	3.30×10^-4	5.428

图1 案例1中N=100，τ=2 s的仿真结果

Fig.1 Simulation results when N=100, τ=2 s for case 1

图2 不同方法的RMSE及计算时间的比较

Fig.2 Comparison of the RMSE and calculation time of different methods

表2 案例2不同网格尺寸的最大绝对误差及收敛阶

Table 2 Maximum absolute error and convergence rate with different grid sizes for case 2

h	HR		HOCD
h	MAE	Rate	MAE	Rate
0.04	18.309	—	2.598	—
0.02	8.004	1.194	0.150	4.114
0.01	2.934	1.448	8.354×10^-3	4.167
0.005	0.921	1.672	6.791×10^-4	3.621

图3 案例2中N=100，τ=2 s的仿真结果

Fig.3 Simulation results when N=100, τ=2 s for case 2

图4 不同方法的RMSE及计算时间的比较

Fig.4 Comparison of the RMSE and calculation time of different methods

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