# relaxation parameters in MRT-advection-dispersion model!

Dear LB members;
I recently started using MRT-model for fluid flow and mass transport in porous media at pore and macroscopic scale.

I am trying to implement D2Q9-MRT-based anisotropic-advection dispersion (AADE) model for mass transport as proposed by Ginzburg (2005, Adv. WR).

I am linking MRT-based flow model (parameters used from Lallemand and Luo, 2000, PRE) with above-mentioned AADE- model. The AADE model works fine as long as relaxation parameters (s1(4,6)) remain less than 1.96 for any flow rate.

I need help to find better set of relaxation parameters (s1) to simulate low-diffusion problem
( (s1(4,6)~1.999) at high flow rate (~0.1 lu/ts).

I can send out my code written in MATLAB for debugging purposes. Following are parameters value I am using.

relaxation parameters:

s1=[0,-1.8,-1.2,-1.96,-1.5,-1.96,-1.5,-1.3,-1.3];
s1(4,6) controls the mass diffusion or Peclet number.

a21=-2;a31=1;c11=0;axx=0;axy= 0;

Following are moments

m_eq1(1) = rho1(j,i); %density
m_eq1(2) = (a21)rho1(j,i); %energy
m_eq1(3) = (a31)rho1(j,i); %energy square
m_eq1(4) = jx
rho1(j,i); %momentum in x-dir
m_eq1(5) = (1/2)c11jx; %energy flux in x-dir
m_eq1(6) = jy
rho1(j,i); %momentum in y-dir
m_eq1(7) = (1/2)c11jy; %energy flux in y-dir
m_eq1(8) = axxrho1(j,i); %diagonal comp of stress tensor
m_eq1(9) = axy
rho1(j,i); %off diagonal comp of stress tensor

microscopic velocities:
e(1,:)=[1,0,-1,0,1,-1,-1,1,0]; %ex
e(2,:)=[0,1,0,-1,1,1,-1,-1,0]; %ey

weights:
wa(9)=4/9;wa(1:4)=1/9;wa(5:8)=1/36;%weights

M-matrix

M=[ 1,1,1,1,1,1,1,1,1;
-1,-1,-1,-1,2,2,2,2,-4;
-2,-2,-2,-2,1,1,1,1,4;
1,0,-1,0,1,-1,-1,1,0;
-2,0,2,0,1,-1,-1,1,0;
0,1,0,-1,1,1,-1,-1,0;
0,-2,0,2,1,1,-1,-1,0;
1,-1,1,-1,0,0,0,0,0;
0,0,0,0,1,-1,1,-1,0;
];

Thanks for help and suggestions.

i felt like using mrt for aade was mainly for the purpose of accuracy and not stability, but I might be wrong. what happens if you use bgk instead of mrt? is the simulation really less stable? and, you could try to increase the stability by using a different boundary condition (but i don’t know which one to use).

MRT is not only more accurate but also lot more stable than BGK for AADE as well as flow simulation. BGK breaks down for small diffusion at high flow rate but MRT worked fine, I had to change a21, which is a free parameter in aade model to improve stability.

Dr. Ginzburg right now is working on that stability analysis and it’s not ready yet. Probably you can contact her directly and ask for better results, or wait a few months until she comes with good results. Right now she obtained some results for TRT but not MRT, if you are interested in it.

Alex

Hello,
Dear members,