Linearization and convergence
The flow solver uses linearization and convergence to solve for pressure, velocity, enthalpy, and other variables. It is important to understand the time versus non-linear convergence loops, and the equations that are solved.
The equations solved by the flow solver alone or when coupled with the thermal solver can be written in the general form of a non-linear partial differential equation, which is defined as:
This general equation is linearized and discretized into the following general form:
- T is the transient term dependent on the variable ϕ value of the previous time (superscript old) and the current linearization iteration (superscript n+1).
- L is all the linear terms including the linear source terms dependent on the variable ϕ value of the current linearization iteration (superscript n+1).
- N is all the non-linear terms including the non-linear source terms dependent on the variable ϕ value of the current and previous linearization iterations (superscripts n+1 and n respectively).
- S is the remaining source terms that are not dependent on the variable ϕ.
Time marching loop
For both steady-state and transient simulations, the flow solver starts with an outer iteration over time as shown in the following pseudo-code.
ϕold = ϕinit
Start time marching loop
Linearize the previous equation and solve for ϕn+1
Update ϕold = ϕn+1
Check for steady-state convergence or the completion of a transient simulation
If yes: exit time marching loop
End of time marching loop
Linearization loop
To resolve for linearization of the values of ϕn+1 versus ϕn, a general form of linearization loop must be added. The purpose of the loop is to ensure that the values of ϕn+1 and ϕn are close enough that they are interchangeable in the non-linear term N(ϕn+1,ϕn) by the end of the loop with little impact on its value, as shown in the following pseudo-code.
ϕold = ϕinit
Start time marching loop
n = 0
ϕ0 = ϕold
Start linearization loop
Solve the previous equation for ϕn+1
Check for non-linear convergence
If yes: exit the linearization loop
Else: update ϕn = ϕn+1 and increment n
End linearization loop
Update ϕold = ϕn+1 and increment n
Check for steady-state convergence or the completion of a transient simulation
If yes: exit time marching loop
End of time marching loop