Calculate the characteristic derivatives for outgoing characteristics using one-sided derivatives and for incoming characteristics from the physical boundary conditions. Then use the boundary equations (49-52 or 53-56) to determine the time derivatives of the variables. Assume subsonic flow. Then there is always one outgoing wave, one incoming wave and the advection (entropy) wave which switches between incoming and outgoing depending on the normal fluid velocity.
Evaluate using one-sided derivatives,
and the equation for the entropy is
For inflows, to keep the entropy constant, set . To make the entropy tend toward some reference value, , set
where the constant for some time constant , which might be taken as or for instance, and the other variables have their mean values or initial boundary values.
For outflows, evaluate using one-sided derivatives,
The no-reflection condition is . However, additional constraints should be imposed to ensure constant bottom pressure or zero net mass flux. Thus take
this should make the net mass flux . Also,
so this should make the pressure .
The characteristic equation for the horizontal velocities is
For outflows evaluate using one-sided derivatives,
For inflows, set
Then calculate the time-derivative of the horizontal velocities from the boundary equations.
Set the normal derivative of the tangential viscous stresses and energy flux to zero,
Finally, solve equations (53-56) for the time-derivatives of the conserved variables or equations (49-52) for the time-derivatives of the primitive variables.