Solver for system of hyperbolic equations
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: http://www.cfd-online.com/Forums/ope...-equation.html) 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?
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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