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## Solution of the discrete Boltzmann equation: Based on the finite volume method

Sun Jia-Kun, Lin Chuan-Dong, Su Xian-Li, Tan Zhi-Cheng, Chen Ya-Lou, Ming Ping-Jian
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• #### 摘要

近十年来, 离散Boltzmann方法在复杂非平衡流体系统领域的应用取得了显著的进展, 这种方法已逐步成为描述和预测流体系统行为的重要手段. 该方法的控制方程是一套简单统一的离散Boltzmann方程, 其离散格式的选取对于数值模拟的计算精度和稳定性有着直接影响. 为了提高数值模拟的可靠性, 本文引入有限体积法用于求解离散Boltzmann方程. 有限体积法是一种常用的数值计算方法, 具有守恒性强、精度高等特点, 可用于有效处理高速可压缩流体的数值计算问题. 本文采用MUSCL格式进行重构, 并引入了通量限制器以提高数值计算的稳定性. 最后, 对基于有限体积的离散Boltzmann方法进行了验证, 数值算例包括冲击波、Lax激波管以及声波. 结果表明, 该方法能够准确刻画冲击波、稀疏波、声波, 以及物质界面的演化, 同时确保系统的质量、动量和能量守恒, 还可以准确测量流体系统的流体力学和热力学非平衡效应.

#### Abstract

Mesoscopic methods serve as a pivotal link between the macroscopic and microscopic scales, offering a potent solution to the challenge of balancing physical accuracy with computational efficiency. Over the past decade, significant progress has been made in the application of the discrete Boltzmann method (DBM), which is a mesoscopic method based on a fundamental equation of nonequilibrium statistical physics (i.e., the Boltzmann equation), in the field of nonequilibrium fluid systems. The DBM has gradually become an important tool for describing and predicting the behavior of complex fluid systems. The governing equations comprise a set of straightforward and unified discrete Boltzmann equations, and the choice of their discrete format significantly influences the computational accuracy and stability of numerical simulations. In a bid to bolster the reliability of these simulations, this paper utilizes the finite volume method as a solution for handling the discrete Boltzmann equations. The finite volume method stands out as a widely employed numerical computation technique, known for its robust conservation properties and high level of accuracy. It excels notably in tackling numerical computations associated with high-speed compressible fluids. For the finite volume method, the value of each control volume corresponds to a specific physical quantity, which makes the physical connotation clear and the derivation process intuitive. Moreover, through the adoption of suitable numerical formats, the finite volume method can effectively minimize numerical oscillations and exhibit strong numerical stability, thus ensuring the reliability of computational results. Particularly, the MUSCL format where a flux limiter is introduced to improve the numerical robustness is adopted for the reconstruction in this paper. Ultimately, the DBM utilizing the finite volume method is rigorously validated to assess its proficiency in addressing flow issues characterized by pronounced discontinuities. The numerical experiments encompass scenarios involving shock waves, Lax shock tubes, and acoustic waves. The results demonstrate the method's precise depiction of shock wave evolution, rarefaction waves, acoustic phenomena, and material interfaces. Furthermore, it ensures the conservation of mass, momentum, and energy within the system, as well as accurately measures the hydrodynamic and thermodynamic nonequilibrium effects of the fluid system. Compared with the finite difference method, the finite volume method is also more convenient and flexible in dealing with boundary conditions of different geometries, and can be adapted to a variety of systems with complex boundary conditions. Consequently, the finite volume method further broadens the scope of DBM in practical applications.

#### 作者及机构信息

###### 通信作者: 林传栋, linchd3@mail.sysu.edu.cn
• 基金项目: 国家自然科学基金(批准号: 51806116)、广东省基础与应用基础研究基金(批准号: 2022A1515012116, 2024A1515010927)和国家留学基金管理委员会(批准号: 202306380288)资助的课题.

#### Authors and contacts

###### Corresponding author: Lin Chuan-Dong, linchd3@mail.sysu.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51806116), the Guangdong Basic and Applied Basic Research Foundation, China (Grant Nos. 2022A1515012116, 2024A1515010927), and the China Scholarship Council (Grant No. 202306380288).

#### 施引文献

• 图 1  离散速度(a)与控制单元的示意图(b)

Fig. 1.  Sketches of discrete velocities (a) and control volumes (b).

图 2  网格无关性验证　(a)不同网格数下冲击波波阵面附近的压强分布; (b)不同空间步长下的相对误差

Fig. 2.  Grid-independence validation: (a) Pressure distribution in the vicinity of the shock wave for different grid numbers; (b) relative errors at different spatial steps.

图 3  冲击波周围的物理量, t = 0.375 　(a) 密度; (b) 水平速度; (c) 温度; (d) 压强

Fig. 3.  Physical quantities around the shock wave, t = 0.375: (a) Density; (b) horizontal velocity; (c) temperature; (d) pressure.

图 4  冲击波周围的非平衡量, t =0.375

Fig. 4.  Nonequilibrium quantities around the shock wave, t =0.375.

图 5  Lax激波管中的物理量, t = 0.15　(a) 密度; (b) 水平速度; (c) 温度; (d) 压强

Fig. 5.  Physical quantities in the Lax shock tube, t = 0.15: (a) Density; (b) horizontal velocity; (c) temperature; (d) pressure.

图 6  在声波传播过程中不同时刻的压强分布图, $\gamma = 1.4,\;T=1.0$, t = 0, 0.050, 0.125, 0.150, 0.175和0.200

Fig. 6.  Pressure contours in the evolution of a sound wave at time instants t = 0, 0.050, 0.125, 0.150, 0.175, and 0.200, respectively, $\gamma = 1.4,\;T=1.0$.

图 7  质量、动量和能量的守恒性验证: 正方形、菱形、三角形和圆形分别表示平均密度、x方向平均动量、y方向平均动量和平均能量. 实线代表对应的精确解

Fig. 7.  Verification of the conservation of mass, momentum and energy. Squares, diamonds, triangles and circles represent the average values of density, momentum in the x direction, momentum in the y direction and energy, respectively. The solid lines denotes the corresponding exact solutions.

图 8  声波的传播　(a)$\gamma = 1.4$, 不同温度; (b)$T = 1.0$, 不同比热比

Fig. 8.  Propagation of the sound wave: (a) $\gamma = 1.4$ with various specific heat ratios; (b) $T = 1.0$ with various temperatures.