# Heat Diffusion Problem Boundary Condition

Hello, I’m an u-grad students and new to Lattice-Boltzmann.

I start learning LBM from simple heat diffusion problem. There was no problem when solving 1D problem. But when I advance to 2D problem, I got a little insecure with my boundary condition.

My system is a 2D plate connected to heat reservoir with temperature Tw, Tn, Te, Ts for west side, north side, east side, and south side respectively.I used D2Q4 model with lattice numbering; (1,2,3,4) for (right, up, left, down) direction.

This is the boundary condition I use;
f1(west) = 0.25Tw;
f2(west) = 0.25
Tw;
f3(west)= 0.25Tw;
f4(west) = 0.25
Tw;
f1(east) = 0.25Te;
f2(east) = 0.25
Te;
f3(east) = 0.25Te;
f4(east) = 0.25
Te;
f1(south) = 0.25Ts;
f2(south) = 0.25
Ts;
f3(south) = 0.25Ts;
f4(south) = 0.25
Ts;
f1(north) = 0.25Tn;
f2(north) = 0.25
Tn;
f3(north) = 0.25Tn;
f4(north) = 0.25
Tn;

I developed this boundary condition myself because I couldn’t find any reference. Is this the correct way to set the boundary condition? However when I compared the result with Finite Difference, the MSE is about 0.1024.

Thank you

Regards,