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Old   August 2, 2001, 18:28
Default boundary condition @ interface
  #1
Steve
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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?
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Old   August 2, 2001, 18:37
Default Re: boundary condition @ interface
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Peter
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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
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Old   August 3, 2001, 08:34
Default Re: boundary condition @ interface
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Doug
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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.
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Old   August 3, 2001, 12:31
Default Re: boundary condition @ interface
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Steve
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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
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Old   August 3, 2001, 12:36
Default Re: boundary condition @ interface
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Steve
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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
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