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?