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dirichelet boundary conditions for hyperbolic problem

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Old   August 9, 2015, 05:59
Cool dirichelet boundary conditions for hyperbolic problem
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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 up-winding but still have the same problem. Please help if this is your area of expertise.

Thank you in advance
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Old   August 9, 2015, 07:05
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Quote:
Originally Posted by Hooman View Post
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 up-winding but still have the same problem. Please help if this is your area of expertise.

Thank you in advance

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
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Old   August 9, 2015, 07:14
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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)
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Old   August 9, 2015, 07:31
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Quote:
Originally Posted by Hooman View Post
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)

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
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Old   August 12, 2015, 17:26
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Thank you.
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