Hi,

Consider the following equation:

a*dF/dt= d/dx[b*dF/dx]

I am trying to solve the above equation using Implicit Finite Diff. Approx. The discretized Eq. takes the form:

a(i,n)*F(i,n+1)-F(i,n)/dt = b(i-1/2,n)*(F(i-1,n+1)-F(i,n+1))/dx + b(i+1/2,n)*(F(i+1,n+1)-F(i,n+1))/dx

i=grid point, n=time level. This equation is to be solved with a phase change region inside the computational domain. F is continous in whole domain (that's why we don't need to determine the different regions). However, a=a(F) and b=b(F) are variable coefficients with high discontinuities as we move from region to another. One of the B.C.s is known and the other one takes the form: b*dF/dx=alfa*F+beta discretized to: b(m,n)*(F(m+1,n+1)-F(m-1,n+1))/2dx=alfa*F(m,n+1)+beta where m=maximum grid pt. I substituted the BC into the Discretized Eq. and then solved the TDMA system. I got non-physical oscillatory solution!!! When I wrote the simultaneous eqs. I found out that for the Boundary grid I have a coefficient (b) outside the comptational domain. I approximate it like: b(i+1/2)~b(i) with no luck. I then approximate it with b(i+1/2) with a fictitous point, I also get non-physical osc. sol. Although I am solving implicit scheme, I am using a small dt. Any suggestions?

Time step must be kept usually small even if you need to perform many iteration. So that shouldn't be the problem. Usually oscilations are due to odd-even decoupling of cells when using CDS scheme, but I see you are not using it. Check out how you are defining in your code the first derivatives

Your message notes you are using a central differece at the boundary. If you are having strong gradients at the boundary this could lead to numerical oscillations.

Try using a first-order upwind difference or adding a bit of artificial dissipation and see if that will help stabilize the solution.

Dear Peter,

Thanks for your reply. I am using FTCS as it appears in my first message. The only first derivative I have is for the unsteady term.

Regards

Dear Doug,

Thanks for your suggestions. I've already tried a backward differencing @ the boundary and I got (numerical) oscillations too. However, I haven't tried to add an artificial dissipation yet and I don't know how to do it. Any references? Note that I am having 1st derivative in time and 2nd derivative in space with nonlinear coefficients!

Regards