# Solver for system of hyperbolic equations

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October 17, 2012, 04:29
Solver for system of hyperbolic equations
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Join Date: Mar 2012
Location: Munich, Germany
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Hi,

I derived the adjoint equations from the twodimensional Eulerequation (with boundary conditions) in theory. Now I would like to solve these with OpenFOAM. Itīs a system of hyperbolic equations, so the normal discretisation of OF doesnīt work (Iīve tried ).

The system is given by

,
,
,
.

where the indices x and y mean the x and y derivation and the aīs and bīs are factors, dependent from the solution of the Eulerequation.

According to the solution of the onedimensional advection equation (you can have a look at this thread: 1D advection equation) I tried to use a special discretisation with interpolation to the cell faces. You can find the code at the bottom.

For the solution I use the same timesteps like for the solution of the Eulerequation, but the adjoint solver is unstable and shows oscillations. My problem is, that I canīt explane me why .

Do you have any suggestions for a correct implementation for this system of hyperbolic equations or some hints, why my solver doesnīt work?

Thank you!

regards

treima
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 October 21, 2012, 09:34 #2 Super Moderator     Praveen. C Join Date: Mar 2009 Location: Bangalore Posts: 244 Blog Entries: 6 Rep Power: 9 You need to use an upwind scheme or add some dissipation. In the scalar case, you are using a central scheme with implicit time scheme. This is L2 stable so the computations do not blow up. But you do not get monotone solution, your solution is not between [0,1] but is going out of this interval.

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