extremely simple problem... can you solve it properly?

Mikhail

Hey there.

While doing CFD, we mostly trying to solve (or even solve!) an extremely complex set of equations, e.g. NS equations. "Fortunately", there are no analytical solutions to them, and for that reason who cares how good numerical solutions are... But there are still the simplest problems that possess exact solutions. Consider the easiest transient problem that I could think of. Here we go... Given is the 1d problem of transient heat diffusion, say, with the following boundary conditions: BC: x=0, T(0,t)=1; x=1, T(1,t)=0; and initial temperature is as follows: T(x,0)=0 An analytical solution to the problem is provided by Carslaw and Jaeger, "Conduction of Heat in Solids", chapter 3.4.

I'm sure every numerical approach will provide us with approximately 10% of error w.r.t. to temperature field for the very first time step, no matter how small time steps are used (I tried FDM, FEM, BEM...).

So, I'm challenging everybody in this NG to numerically solve this simplest heat diffusion problem. You may use any commercial, or your own codes -- please provide here a feedback of your experience.

The bottom line here is: if we fail to get any better accuracy than 10% for the initial time step even for LINEAR problem, what kind of numerical results are we getting for more complex NON-linear problems governed by NS equations? So what if the whole problem is very sensitive to small perturbances, but we introduce error that is no longer "small"?

PS of course, not for every fluid flow problem do we get such a tough boundary condition, but let's see what happens... Thank you for your response. Mikhail