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In the book Discontinuous Galerkin Method (DGM), Analysis and Applications to Compressible Flow by Vít Dolejší and Miloslav Feistauer, Springer, it is mentionned, in section 8.3.2 that deals with inviscid compressible flows, that imposing the inlet/outlet conditions as Dirichlet conditions in not enough since "since [the] system is hyperbolic". Then the authors proceed to show an "Approach Based on the Solution of the Linearized Riemann Problem".

I have two questions:

  • Why not simply impose inlet/outlet conditions as Dirichlet conditions? The idea that comes to my mind is that we process the fluxes to further impose the boundary conditions in addition to ghost cells having the right values.
  • More generally, how are inlet/outlet conditions imposed in the DGM when solving the Navier-Stokes equations?
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Your question has nothing to do with DG in the first place.

Anyway, recall that for the compressible Euler equations following eigenvalues are important for the information propagation

$\lambda_{1}=|\mathbf{v}| + c, \quad \lambda_{2,3,4}=|\mathbf{v}|, \quad \lambda_{5}=|\mathbf{v}| - c.$

Depending of the physical situation at the boundary, some characteristics may leave and others may enter the domain. Simply choosing Dirichlet boundaries for all conservative state variables

$ \begin{align} \mathbf{U}= \left[\rho,~ \rho u,~\rho v,~\rho w,~\rho e\right]^{\mathrm{T}}, \end{align} $

would violate the correct propagation of information. For the compressible Navier-Stokes equations you additionally have to take into account spatial gradients

$ \begin{align} \frac{\partial T}{\partial \vec{n}},~ \frac{\partial u}{\partial \vec{n}},~ \frac{\partial v}{\partial \vec{n}},~ \frac{\partial w}{\partial \vec{n}}, \end{align} $

which are not completely free.

Depending on what you actually want to achieve + considering the number of entering/leaving characteristics, you have different options.

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  • $\begingroup$ Does this remark for any type of boundary conditions when solving Euler/NVS equations? $\endgroup$
    – L Maxime
    Commented Jul 4 at 16:55
  • $\begingroup$ The choice of a specific BC for inlet or outlet simply depends on the physical situation. Considering an outlet, if your flow has a velocity less then the highest characteristic speed you may apply a subsonic BC which enables you to specify a reference pressure, all other values must be taken from inside of the domain. If your flow is faster than any characteristic speed you can choose all values from inside. This however won't work if you are interested in accoustic simulations. Here you may rather use non-reflective BCs. You won't get a specific answer without providing more informations. $\endgroup$
    – ConvexHull
    Commented Jul 5 at 5:04
  • $\begingroup$ Do you need to solve for characteristics of the linearized Riemann problem to impose any boundary condition for such equations? According to the book, this is an approach that doesn't exploit the properties of the flows. Another method is the one u are mentioning. I do not have a particular flow in mind. $\endgroup$
    – L Maxime
    Commented Jul 5 at 7:19
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    $\begingroup$ No, you don't have to. $\endgroup$
    – ConvexHull
    Commented Jul 7 at 14:44

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