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dirichelet boundary conditions for hyperbolic problem 

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August 9, 2015, 05:59 
dirichelet boundary conditions for hyperbolic problem

#1 
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Hi,
I am solving a set of 1D hyperbolic equation for a variable density flow. So far I have been using averaging to discretise p and v at the boundary. At the boundaries I have been using linear extrapolation to p and v at the left and right boundaries. For instance, for velocity at the left boundary I have used vw = v(1)  2*v(2) where v(1) and v(2) represent the first and second nodes from the left boundary. To impose the Dirichlet boundary conditions, I have either just set vw = BC or vw = BC  2*v(2) I think this is what is the virtual node approach (?) The problem I have with this method is that as I increase my number of nodes I get alot of fluctuations as if a wave reaches the boundary and is reflected back. I am looking for an easy fix to the problem. I tried using upwinding but still have the same problem. Please help if this is your area of expertise. Thank you in advance 

August 9, 2015, 07:05 

#2  
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Filippo Maria Denaro
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Quote:
If you have Dirichlet B.C.s you need just to set values at one boundary, in your case the left one. I don't understand why you want to use extrapolation...no ghost points are needed 

August 9, 2015, 07:14 

#3 
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Hi,
Thank you, that is what I had done originally but it caused fluctuations at the boundary. Therefore, after searching literature I found virtual nodes, and like you said it did not help. If I did not have a Dirichlet BC, and I just needed to define the left boundary would it be correct to use the extrapolation? I am in particular, having problems with my pressure BC, I have come across some papers where they simply use pw = p(1) or pe = p(end) 

August 9, 2015, 07:31 

#4  
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Filippo Maria Denaro
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no, for hyperbolic equations the BC.s are dictated by the direction of the characteristic lines. Thus, extrapolation at the left boundary can be used only if you have a wave coming from the interior and going toward the left boundary. If you have Euler equation and subsonic flows, you have 2 characteristics coming from the left boundary and going into the interior and 2 Dirichlet conditions are required, 1 characteristic is from the interior and you can use extrapolation for that condition. If your flow is supersonic the 3 characteristics require 3 Dirichlet BC.s 

August 12, 2015, 17:26 

#5 
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Thank you.


Tags 
boundary condition, dirichlet, discretization, extrapolation, hyperbolic 
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