压力修正算法对刮削层等离子体输运方程数值求解性能的影响

doi:10.19596/j.cnki.1001-246x.8525

摘要/Abstract

摘要：

基于C++撰写测试程序，研究SIMPLE(Semi-Implicit Method for Pressure Linked Equations)、SIMPLEC(SIMPLE Consistent)、SIMPLER(SIMPLE Revised)、SIMPLEX(SIMPLE Extrapolation)以及PISO(Pressure-Implicit with Splitting of Operators)等压力修正算法对刮削层等离子体Braginskii输运方程数值求解性能的影响。测试程序采用SOLPS(Scrape-Off Layer Plasma Simulation)的等离子体模型方程，数值计算针对简化平板模型开展。结果表明: 5种压力修正算法均能使程序收敛至正确的结果，其中PISO算法的收敛速度最快，SIMPLE、SIMPLEC、SIMPLER、SIMPLEX的收敛速度无明显差异。

关键词: 托卡马克, 刮削层, Braginskii方程, 有限体积法, 压力修正算法

Abstract:

Test codes are programmed with C++ to study influence of pressure correction algorithms, including SIMPLE (Semi-Implicit Method for Pressure Linked Equations), SIMPLEC (SIMPLE Consistent), SIMPLER (SIMPLE Revised), SIMPLEX (SIMPLE Extrapolation) and PISO (Pressure-Implicit with Splitting of Operators), on numerical solution performance of Braginskii transport equation for scrape-off layer plasma. Plasma model equations of SOLPS (Scrape-Off Layer Plasma Simulation) are adopted in the test codes. Numerical calculations are carried out on a simplified slab model. It was found that all of the pressure correction algorithms make the program converge to correct results. The fastest convergence speed is achieved with PISO. There is no significant difference in the convergence speed of SIMPLE, SIMPLEC, SIMPLER and SIMPLEX algorithms.

Key words: tokamak, scrape-off layer, Braginskii equation, finite volume method, pressure correction algorithm

朱仁柱, 何家丰, 黄泰豪, 阮行磊, 徐宇晨, 郭晋, 刘天元, 毛世峰, 叶民友. 压力修正算法对刮削层等离子体输运方程数值求解性能的影响[J]. 计算物理, 2023, 40(1): 29-39.

Renzhu ZHU, Jiafeng HE, Taihao HUANG, Xinglei RUAN, Yuchen XU, Jin GUO, Tianyuan LIU, Shifeng MAO, Minyou YE. Performance of Numerical Calculation of Transport Equations of Scrape-off Layer Plasma: Pressure Correction Algorithms[J]. Chinese Journal of Computational Physics, 2023, 40(1): 29-39.

图/表 8

图1 简化平板模型示意图

Fig.1 A schematic of the simplified slab model

表1 边界条件设置

Table 1 Boundary condition settings

	密度/m^-3	平行速度	电子温度/eV	离子温度/eV
芯部(图 1中蓝色边界)	3×10¹⁹	$ \partial V_{\\|} / \partial y=0$	400	400
靶板(图 1中黄色边界)	$ \partial n / \partial x=0$	±6.189 938×10⁴ m·s^-1*	40	40
私有通量区/刮削层(图 1中红色边界)	3×10¹⁸	$ \partial V_{\\|} / \partial y=0$	40	40

图2 使用SIMPLE算法在细网格计算得到的(a)离子密度、(b)离子平行速度、(c)电子温度、(d)离子温度分布以及(e)压力分布

Fig.2 Distributions of (a) ion density, (b) ion parallel velocity, (c) electron temperature, (d) ion temperature and (e) pressure calculated with SIMPLE algorithm on fine grid

图3 使用SIMPLE算法在不同密度网格上计算得到的在y=-0.005 m处的(a)离子密度、(b)离子平行速度、(c)电子温度和(d)离子温度分布的极向分布与Richardson外推结果(紫色实线)

Fig.3 Poloidal distributions of (a) ion density, (b) ion parallel velocity, (c) electron temperature and (d) ion temperature at y=-0.005 m calculated with SIMPLE algorithm and Richardson extrapolation (solid purple line)

图4 用5种不同压力修正算法计算(a)离子密度、(b)离子平行速度、(c)电子温度和(d)离子温度极向分布的外推结果

Fig.4 Extrapolated results of (a) ion density, (b) ion parallel velocity, (c) electron temperature and (d) ion temperature with five pressure correction algorithms

图5 压力松弛因子分别为(a) 0.3、(b) 0.5、(c) 0.7、(d) 0.9，不同压力修正算法得到的密度残差随迭代次数的变化

Fig.5 Density residuals calculated with different pressure correction algorithms as functions of the number of iterations, while the pressure relaxation factors are (a) 0.3, (b) 0.5, (c) 0.7, and (d) 0.9, respectively

图6 压力松弛因子分别为(a) 0.3、(b) 0.5、(c) 0.7、(d) 0.9，用不同压力修正算法得到的平行速度残差随迭代次数的变化

Fig.6 Parallel velocity residuals calculated with different pressure correction algorithms as function of the number of iterations, while the pressure relaxation factors are (a) 0.3, (b) 0.5, (c) 0.7, and (d) 0.9, respectively

表2 使用不同压力修正算法的收敛速度(压力松弛因子为0.5。)

Table 2 Convergence velocity using different pressure correction algorithms (Pressure relaxation factor is 0.5.)

算法	单轮计算时间/s	收敛迭代次数(密度残差低于10¹⁴)	计算收敛时间/s
SIMPLE	0.766	823	630.4
SIMPLEC	0.744	822	611.6
SIMPLER	0.823	822	676.5
SIMPLEX	0.795	825	655.9
PISO	0.813	466	378.9

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