Optimal computational parameters for maximum accuracy and minimum cost of Arnoldi-based time-stepping methods for flow global stability analysis

Abstract

Global instability analysis of flows is often performed via time-stepping methods, based on the Arnoldi algorithm. When setting up these methods, several computational parameters must be chosen, which affect intrinsic errors of the procedure, such as the truncation errors, the discretization error of the flow solver, the error associated with the nonlinear terms of the Navier-Stokes equations and the error associated with the limited size of the approximation of the Jacobian matrix. This paper develops theoretical equations for the estimation of optimal balance between accuracy and cost for each case. The 2D open cavity flow is used both for explaining the effect of the parameters on the accuracy and the cost of the solution, and for verifying the quality of the predictions. The equations demonstrate the impact of each parameter on the quality of the solution. For example, if higher order methods are used for approaching a Fr\'echet derivative in the procedure, it is shown that the solution deteriorates more rapidly for larger grids or less accurate flow solvers. On the other hand, lower order approximations are more sensitive to the initial disturbance magnitude. Nevertheless, for accurate flow solvers and moderate grid dimensions, first order Fr\'echet derivative approximation with optimal computational parameters can provide 5 decimal place accurate eigenvalues. It is further shown that optimal parameters based on accuracy tend to also lead to the most cost-effective solution. The predictive equations, guidelines and conclusions are general, and, in principle, applicable to any flow, including 3D ones.