请教一个MRT-D3Q19构建的过程。

lxp121

各位，我一直对MRT的构建过程不是很清楚，特别是如何构建转换矩阵M和平衡矩M^eq，主要有如下几个问题，请大家指教一下：

1. D3Q19中宏观量（矩）的选择，
首先是4个守恒量：\rho, \rho u_x, \rho u_y 和 \rho u_z;  对应 |e |^0,  ex, ey, ez;

然后是二阶项：e, p_{xx}, p_{xy}, p_{xz}, p_{yz}, p_{ww}; 对应 |e|^2, ex^2, e_{xy}, e_{xz}, e_{yz} 和 ey^2 - ez^2;

但是对于三阶矩（j_x, j_y, j_z, m_x, m_y, m_z ) 是如何选择的？对应了哪些三阶矩？
四阶矩（\epsilon, \pi_{ww}, \pi_{xx}, m_x, m_y, m_z ）是如何选择的呢？对应哪些四阶矩？

2. 关于Gram-Schmidt过程，选定了上面这些矩之后，应该对他们进行正交化。
这个时候正交化的顺序会改变最后的矩阵顺序，为什么现有的MRT中这些矩的排列顺序是这样？
另外转换矩阵M为什么一定需要正交化？

3. 对应于转换矩阵M，如果确定各个平衡矩，特别是高阶矩的确定。

4. 另外，对于作用力的添加是否可以直接根据其对各个宏观量的影响直接在矩空间中实现，而不必通过Guo提出的 F = M^{-1} ( I - .5 S ) M F'得到。

很多文献里面对这个过程描述比较简略，也可能是我看得不仔细，不到位，请各位指导一下，谢谢。

luo@odu.edu

1) you can start with the mononomials  (c_x^n)*(c_y^m)*(c_z^k), Gram-Schmid procedure tells which ones are LINEARLY independent ones and you have no choice but to use them -- for a model with Q discrete velocities, you will only find Q LINEARLY independent  moments.

2) The ordering (or labeling) of the moments is arbitrary, you could use whatever order you prefer. It does NOT change anything. We use the order according to the meaning of the moments, and in their order, too.

If you do not orthogonalize the rows of M, the moments will have "mixed" effect, i.e., you cannot adjust each moment individually -- they are all interconnected.

3) Once you know the polynomials which correspond to each moments, they tells you the equilibria -- (c_x^n)*(c_y^m)*(c_z^k) is translated to (u_x^n)*(u_y^m)*(u_z^k). Of course some higher order terms can be neglected.

4) Yes -- you are correct. This is very good reason to use MRT.

-- LS Luo

[ 本帖最后由 luo@odu.edu 于 2012-9-13 04:24 编辑 ]

bqhrz

关于平衡矩的确定还是不清楚，例如：
MRT-D2Q9中，矩e对应的多项式为|e>=-4|e|^0+3|e|^2，为什么其平衡函数为
e_eq=ρ(-2|u|^0+3|u|^2)，而不是e_eq=ρ(-4|u|^0+3|u|^2)呢？

[ 本帖最后由 bqhrz 于 2012-8-25 09:41 编辑 ]

luo@odu.edu

You have to take care of the "normalizations" of the moments. See the formulas (16a) -- (16f) in the paper by Lallemand & Luo [Phys. Rev. E 61(6):6546 (2003)], which can be downloaded from my website (http://www.lions.odu.edu/~lluo)if you do not have the access to the paper.