Shallow Water Equations Boundary Conditions

Question

I am trying to solve shallow water equations using DG methods. Flow over a bump is a common problem that comes up in this context. For example (http://loki.udc.es/info/asignaturas/calculo_ii/Finite%20Volumes%20IMWE.pdf, page 46). These are my questions:

1. The boundary conditions are mentioned only for the upstream discharge and downstream height. How can I get the boundary values for upstream height and downstream discharge that is required for flux calculations?
2. How can I apply slope limiters at the boundaries in this case or even a simpler case of an advection equation where Dirichlet boundary condition is specified at the boundary?

Answer 1

1. For hyperbolic PDEs often the specification of the boundary conditions tend to only be needed on one boundary. For a system of equations like you have specified the boundary conditions needed are exactly the ones you mentioned. When implementing these numerically we often specify the other boundary conditions as extrapolated boundary conditions, in the simplest case just copying the values from the domain into the ghost cells.
   2. With specification of ghost cells calculation of the slope should be possible for all the boundaries of cells.