[DubsVeer23]

Hi,
I couldn't find an FDM forum so I decided to post here. I am sure many of you can help me with my problem

It's 1-D heat diffusion with oscillatory boundary condition. A semi-infinite rod has it's one end heated: u(0,t) = sin(t) and the other end: u(L,t) is bounded.
The initial temperature profile is u(x,0) = 0.

I have to obtain temperature profiles at various times using (Euler) implicit FDM scheme.

The heat equation is: u_t = u_xx and the discretization is: u(n+1) = u(n,i) + (k/h^2)(u(n+1, i+1)-2u(n+1,i)+u(n+1,i-1))

I am struggling with the oscillatory boundary condition. Analytically, this condition introduces lag (e^(-x)*sin(t-phi)) in the rod, I can understand that but numerically how does one approach this situation (esp. the boundary condition) so that the analytical solution is well approximated.

Any help/suggestion is appreciated.

[FMDenaro]

This is a homework and you can just see that the BC applies on the node i=1 (x=0) while you solve the equation at the inner node i=2 (x=h).

[DubsVeer23]

Originally Posted by FMDenaro

This is a homework and you can just see that the BC applies on the node i=1 (x=0) while you solve the equation at the inner node i=2 (x=h).

Thanks. That is my approach actually but I am unable to approximate the analytical solution properly. Upon some reading, I understand that the temperature in the rod should lag the surface (boundary temperature), however, I am unable to fathom how a numerical scheme could achieve this. Can you expand your approach or suggest any reading? Thanks!

PS. It's an exercise problem in a book I am reading "Finite Difference Computing with PDEs"