Please, help me
You have to ask specific questions to get help
on the mail list. No one has time to help you
figure out how to create and run a complex simulation.
Did you try to test the obvious? For example, if you plot the temperature vs distance, does it reach the linear shape, and, is the gradient value almost identical to the expected one? Did you check that the heat flux is converged by the time you claim steady state has been reached? Your timestep is quite small thus perhaps the plateau you are seeing does not correspond to the final equilibrium value. Did you check system size effects?..