The characteristic equations (13) are
Write the z-derivatives in the characteristic directions as
For incoming characteristics these are the terms that must be determined from the physical boundary conditions. For outgoing characteristics these terms are evaluated using one-sided derivatives.
The equations at the boundary can be derived by either (i) solving (43) for in terms of and substituting in primitive Eqns. 21-24, or (ii) transforming the characteristic equations (12),
into the primitive equations by multiplying them by to obtain
where the z-derivatives can be found as the solution of the system of equations
since we have the matrix whose rows are the left eigenvectors, but not the matrix .
The primitive z-derivatives in terms of the characteristic derivatives are:
The boundary equations for the primitive variables are obtained by substituting these expressions for the z-derivatives of the primitive variables in the primitive equations (21-24),
where . Hence,
These could also be obtained by solving the system of equations
with an algebra program.
The equations for the quasi-conserved variables are obtained by multiplying Eqns. 49-52 by , which gives
where stands for Eqn. (i).
Thus the quasi-conserved boundary equations are