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- - **Solver for system of hyperbolic equations**
(*https://www.cfd-online.com/Forums/openfoam-programming-development/108202-solver-system-hyperbolic-equations.html*)

Solver for system of hyperbolic equations1 Attachment(s)
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: 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? Thank you! regards treima |

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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