What Is Convergence?

Convergence is a major issue with the use of CFD software. Fluid mechanics is involved with non-linear processes, dealing with inherently unstable phenomena such as turbulence. CFD software is intended to simulate these physical processes, and therefore is subject to the same issues as the processes it is trying to represent. As such it is not guaranteed that there will be a steady-state ‘converged’ solution to a problem.

Computational Fluid Dynamics problems in general are non-linear, and the solution techniques use an iterative process to successively improve a solution, until ‘convergence’ is reached. Convergence  can mean many things to many people. One definition [1 ] is

‘More formally, in mathematics, convergence describes limiting behaviour, particularly of an infinite sequence or series toward some limit . To assert convergence is to claim the existence of a limit, which may be itself unknown. For any fixed standard of accuracy, you can always be sure to be within it, provided you have gone far enough.’

As this definition indicates, the exact solution to the iterative problem is unknown, but you want to be sufficiently close to the solution for a particular required level of accuracy. Convergence therefore does need to be associated with a requirement for a particular level of accuracy. This requirement depends upon the purpose to which the solution will be applied.

Convergence is also often measured by the level of residuals, the amount by which discretised equations are not satisfied, and not by the error in the solution. The user should therefore be aware of this, in deciding what convergence criterion should be used to assess a solution.