From the characteristic equations (repeated below), we can construct three convenient equations known as the compatibility equations.

	(Positive Wave Characteristic)
	(Negative Wave Characteristic)
	(Particle Path Characteristic)

The only unknown terms in these equations are the pressure, velocity and density at point 4. The other terms are found from a previous time step, or from the initial conditions as determined from the steady state MOC simulation.

Rearranging terms:

and defining six new constants (constant for this time step):

We have our simplest equations yet:

Which can be solved directly for pressure and velocity, then used to directly calculate density.

Note:The compatibility equations allow a solution of all interior points in a pipe. However, the pipe endpoints are solved by applying specific boundary conditions. The boundary condition relationships build on the previous methods, and offer the final needed information to generate a complete solution.