Boundary conditions are needed (along with interior information) to determine the time derivatives of the variables, , on the boundaries. A hyperbolic system can be decomposed into wave modes propagating along the characteristic directions. In 1D these directions are unique. In multi-dimensions there are no unique propagation directions because the coefficient matrices ( ) for the different directions k are not simultaneously diagonalizable. However, boundary conditions can be applied one coordinate direction at a time. Outgoing waves are completely determined by the interior solution and no boundary condition can be applied to them. Incoming waves depend on conditions exterior to the computational domain and require boundary conditions. Thus the number of boundary conditions imposed must equal the number of incoming characteristics.
For outgoing waves the characteristic equation can be stably evaluated numerically using one-sided derivatives involving only interior and boundary points. Hence, the boundary condition on outgoing characteristics is the characteristic equation (12 or 13),
evaluated using one-sided derivatives in the -direction.
For incoming waves, the normal derivatives, , must be evaluated using external data to achieve stable equations. The other terms can be evaluated from the interior solution on the boundary. Problems arise at the intersection of boundary surfaces where more than one direction is normal to the boundary.
Kevin Thompson, ``Time-Dependent Boundary Conditions for Hyperbolic Systems, II'', J. Comp. Phys., 89, 439-461 (1990)
T. J. Poinsot and S. K. Lele, ``Boundary Conditions for Direct Simulations of Compressible Viscous Flows'', J. Comp. Phys., 101, 104-129 (1992)