An interpretation and derivation of the lattice Boltzmann method using Strang splitting
An interpretation and derivation of the lattice Boltzmann method using Strang splitting
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DOI：
10.1016/j.camwa.2011.08.047
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发表时间：
2013-01-01
影响因子：
2.9
通讯作者：
Dellar, Paul J.
中科院分区：
文献类型：
--
作者：
Dellar, Paul J.
格子玻尔兹曼空间/时间离散，通常是从沿着特征的积分导出的，被证明对应于解耦流和碰撞步骤之间的斯特朗分裂。斯特朗分裂在两个非交换算子的组合下提供了进化的二阶精确近似，这里用离散玻尔兹曼偏微分方程中的流和碰撞项来标识。奇异分裂通过对称分解实现二阶精度，其中一个算子在半个时间步长中应用两次，另一个算子在整个时间步长中应用一次。我们证明，碰撞半时间步长的自然定义会导致与先前使用不同推理引入的变量相同的变化，以便从离散玻尔兹曼方程沿特征的积分获得二阶准确且显式的方案。这种方法很容易扩展到包括一般矩阵碰撞算子以及体积力。最后，我们表明，对于大于 1 的网格尺度雷诺数，格子玻尔兹曼离散的有效性主要取决于使用克兰克-尼科尔森近似来离散碰撞算子。对于与流分离的碰撞，用现成的精确解决方案替换这种近似会导致由于分裂误差而变得过于扩散的方案，除非网格尺度雷诺数保持远低于统一。 (C) 2013 年由爱思唯尔有限公司出版
The lattice Boltzmann space/time discretisation, as usually derived from integration along characteristics, is shown to correspond to a Strang splitting between decoupled streaming and collision steps. Strang splitting offers a second-order accurate approximation to evolution under the combination of two non-commuting operators, here identified with the streaming and collision terms in the discrete Boltzmann partial differential equation. Strang splitting achieves second-order accuracy through a symmetric decomposition in which one operator is applied twice for half timesteps, and the other operator is applied once for a full timestep. We show that a natural definition of a half timestep of collisions leads to the same change of variables that was previously introduced using different reasoning to obtain a second-order accurate and explicit scheme from an integration of the discrete Boltzmann equation along characteristics. This approach extends easily to include general matrix collision operators, and also body forces. Finally, we show that the validity of the lattice Boltzmann discretisation for grid-scale Reynolds numbers larger than unity depends crucially on the use of a Crank-Nicolson approximation to discretise the collision operator. Replacing this approximation with the readily available exact solution for collisions uncoupled from streaming leads to a scheme that becomes much too diffusive, due to the splitting error, unless the grid-scale Reynolds number remains well below unity. (C) 2013 Published by Elsevier Ltd