On the accuracy of time-stepping methods for flow stability analysis

Resumo

Computing the linear stability modes of a flow requires obtaining the eigenmodes of the Jacobian matrix
of its governing equations. For bi-global and tri-global stability analysis, this matrix is very large, which poses a
challenge for solving its eigenproblem. Time-stepping methods are one of the possible solutions for this problem,
as they do not require explicitly building this matrix. One advantage of this class of methods is their relative ease
of implementation, as it can treat the flow solver as a black box, which is integrated into the rest of the algorithm.
The routine calls the flow solver multiple times with different initial conditions to observe the flow behavior. We
study the trade-off between two of the parameters that must be chosen when setting up a case: The length of time
each call to the flow solver will last and the total number of iterations. Longer call times allow the algorithm
to converge in fewer iterations but each one is more costly. The available literature indicates that for a constant
product of call time by number of calls, the accuracy should be roughly the same. We seek both upper and lower
boundaries for the call time length as well as an optimum value that will generate the most accurate results for a
given computational cost.