变阶数时间分数阶Navier-Stokes方程的格子Boltzmann方法

Lattice Boltzmann Method for Variable-order Time-fractional Navier-Stokes Equation

  • 摘要: 发展了一种求解变阶数时间分数阶Navier-Stokes方程的格子Boltzmann(LB)方法。首先,通过对变阶数分数阶导数的历史部分和当前部分分别进行离散,使得变阶数时间分数阶Navier-Stokes方程转化为整数阶偏微分方程。为了提高计算效率,对分数阶导数项的历史部分,除了采用L1直接离散方法进行离散,还采用快速估计方法进行计算;随后,构建了基于2维9速度(D2Q9)格子模型的LB模型,并开展了Chapman-Enskog分析,确定了平衡态分布函数的具体表达式,并理论证明了该LB模型能够准确推导出目标宏观方程;最后,通过数值算例检验所构建的LB模型能够准确求解变阶数时间分数阶Navier-Stokes方程,并且该LB模型的空间收敛阶数约为二阶。

     

    Abstract: Lattice Boltzmann (LB) method for solving the variable-order time-fractional Navier-Stokes equation is developed. Firstly, by discretizing the historical part and the current part of the variable-order fractional derivative respectively, the variable-order time-fractional Navier-Stokes equation is transformed into an integer-order partial differential equation. To improve the computational efficiency, in addition to using the L1 direct discretization method to discretize the historical part of the fractional derivative term, a fast estimation method is also adopted for calculation. Subsequently, LB model based on the two-dimensional nine-velocity (D2Q9) lattice model is constructed, and the Chapman-Enskog analysis is carried out to determine the specific expression of the equilibrium distribution function. It is theoretically proved that this LB model can accurately derive the target macroscopic equation. Finally, through numerical examples, it is verified that the constructed LB model can accurately solve the variable-order time-fractional Navier-Stokes equation, and the spatial convergence order of this LB model is approximately of the second order.

     

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