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I am trying to understand how boundary conditions are implemented when one uses the nonlinear LU-SGS algorithm for Euler equations. Most papers describe the Gauss-Seidel sweep over mesh cells, but do not explain how are boundary conditions applied. I could either consider the same BC implementation as in explicit case (i.e. sweep over cells, then add flux corrections on boundaries), or add Jacobian contributions due to boundary conditions to the element Jacobians. The latter leads to a more complicated implementation, however, where one is obliged to check for each element whether it is supposed to receive boundary Jacobian contributions. Are you aware of any reference discussing LU-SGS and boundary conditions for compressible Euler/Navier-Stokes equations in more detail?

Do you think it is possible to replace the 'boundary condition Jacobian' approach with something simpler? Would that deteriorate the convergence rate of the solver?

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Check the article in [1].

The $\delta Q_f$ (for a boundary face) must be transformed by a matrix $T$ such that,

$\delta Q_f = T * \delta Q_L$ ( note $L$ is the left cell and for a boundary face there is no right cell or $Q_R$ state)

This matrix $T$ is obtained from the boundary condition itself, for example, a supersonic outlet boundary has $T$ = $I$ (identity matrix) and so on ...

Yes, it gets a bit complicated for the wall, and other subsonic BCs. But it is not very hard to derive them yourself. In reference 1, these matrices are already derived for wall and subsonic BC.

References:

[1] John T. Batina, "Implicit upwind solution algorithms for three-dimensional unstructured meshes", AIAA Journal, Vol. 31, No. 5 (1993), pp. 801-805.

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