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This question is (sort of) a continuation of this previously asked question. I am wondering about, in general, how we construct well-posed boundary conditions (both continous and numerical) for flow of a fluid, in particular for the 1-dimensional SWE:s (shallow water equations), such that they correspond to sub-critical and super-critical flow, respectively. I have made some research and it does not seem like LeVeque adresses this in his book Finite Volume Methods for Hyperbolic Problems (I find some stuff on supersonic and transonic waves for the Euler equations, but these are not the same, although similar). Most papers I have found adress this issue for the 2-dimensional case, which I am starting to think is quite analogous to the 1-dimensional case, but I cannot quite derive the conditions to the 1-dimensional case from there myself.

In the case I am wondering about specifically we model a wave (or a pair of waves) on shallow water using the SWE:s and we want to have inflow and outflow boundary conditions (these will correspond to Dirichlet conditions). We solve this numerically using the Roe solver (an approximate Riemann solver). If we have sub-critical flow, how and where do we set the boundary conditions? And similarily for super-critical flow. Additional question: can there be both sub- and super-critical flow in the solution at the same time?

I feel like this might be a trivial question, but I have not been very successful in finding any material on it (although I have a feeling there is a lot of it out there).

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