5.12 Steadystate solution
The equations in Sec. 5.10 are combined using algorithms that couple the solutions for and . One algorithm is SIMPLE (SemiImplicit Method for PressureLinked Equations)^{3}, which is presented here with a modern interpretation.
The SIMPLE algorithm is generally used to produce steady ﬂow solutions in CFD. These solutions are directly applicable for ﬂows that reach a steady state, i.e. when ﬂow variables stop changing in time. They can also provide approximate solutions to ﬂows that are moderately unsteady, usually at a lower cost than a more exact transient solution.
An example of the algorithm is shown for the system of equations presented in Sec. 5.9 . The time derivative () terms are omitted due to the steadystate assumption.
The algorithm involves an iterative sequence with steps, . It begins by constructing a matrix equation for energy which is underrelaxed by a factor . The equation is solved for , which is used to update according to an equation of state. A matrix equation is then constructed using all the terms from the momentum equation excluding , i.e.

(5.21) 
and are then evaluated from (the momentum matrix), as described on page 351. They are used to form the pressure equation, which is solved for .
The new pressure is used to correct the ﬂux so that it obeys mass conservation better (the ﬂux corrector). It is then underrelaxed by a factor before correcting before the next solution step begins (the momentum corrector).