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

The only unknown terms in these equations are the pressure and mass flow at point P. The other terms are known either from a previous time step, or from the initial conditions as determined from a steady state simulation.

Rearranging terms:

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

We have our simplest equations yet:

(1)

(2)

Which can be solved directly for pressure or mass flow:

(3)

(4)

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.