Deterministic numerical solutions of the Boltzmann equation using the fast spectral method
Deterministic numerical solutions of the Boltzmann equation using the fast spectral method
复制标题
使用快速谱方法求解玻尔兹曼方程的确定性数值解
DOI：
10.1016/j.jcp.2013.05.003
复制
发表时间：
2013
影响因子：
4.1
通讯作者：
Wu L
中科院分区：
文献类型：
--
作者：
Wu L
玻尔兹曼方程描述了稀薄气体流动的动力学，但其碰撞算子的多维性质对其数值解提出了真实的挑战。本文将Mouhot和Pareschi提出的用于碰撞算子数值近似的快速谱方法[36]扩展到处理其他碰撞核，例如与软、Lennard-Jones和刚性吸引势相对应的碰撞核。通过比较空间齐次Boltzmann方程的数值解与麦克斯韦分子气体的Bobylev-Krook-Wu精确解，检验了快速谱方法的准确性.结果表明，在计算核模时，用Gauss-Legendre积分代替梯形法则，提高了计算精度;在不损失谱精度的情况下，用拉格朗日乘子法保证了动量和能量守恒。然后比较了具有相同剪切粘度值的不同碰撞核的弛豫到平衡过程;数值结果表明，只要恢复剪切粘度（不仅是剪切粘度值，而且是其温度依赖性），就可以使用不同形式的碰撞核。采用迭代方法求解空间非齐次Boltzmann方程的定常解，其数值误差呈指数衰减。四个经典的基准问题进行了研究：正常的冲击波，和平面傅立叶/Couette/力驱动Poisonille流。对于正常的冲击波，我们的数值计算结果进行了比较与有限差分解的玻尔兹曼方程的硬球分子，实验数据，和分子动力学模拟氩使用现实的Lennard-Jones潜力。对于平面Fourier/Couette/力驱动Poiffille流，我们的结果进行了比较，与直接模拟Monte Carlo方法。在所有的测试情况下，观察到良好的协议，证明了快速谱方法的优点，作为一个计算效率高的方法稀薄气体动力学。
The Boltzmann equation describes the dynamics of rarefied gas flows, but the multidimensional nature of its collision operator poses a real challenge for its numerical solution. In this paper, the fast spectral method [36], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard–Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical solutions of the space-homogeneous Boltzmann equation with the exact Bobylev–Krook–Wu solutions for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss–Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared; the numerical results indicate that different forms of the collision kernels can be used as long as the shear viscosity (not only the value, but also its temperature dependence) is recovered. An iteration scheme is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation, where the numerical errors decay exponentially. Four classical benchmarking problems are investigated: the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows. For normal shock waves, our numerical results are compared with a finite difference solution of the Boltzmann equation for hard sphere molecules, experimental data, and molecular dynamics simulation of argon using the realistic Lennard–Jones potential. For planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the direct simulation Monte Carlo method. Excellent agreements are observed in all test cases, demonstrating the merit of the fast spectral method as a computationally efficient method for rarefied gas dynamics.