Minimal gain time marching schemes for the construction of accurate steady-states

Abstract

Accurate reference solutions are very important in stability analysis, where they must act as a reliable base-state. They are also quite useful for unsteady numerical simulations, where they play key roles as initial conditions and in the implementation of boundary conditions, such as buffer zones. Quite often they are approximate solutions for a simplified version of the particular problem at hand, such as boundary-layer solutions. However, these approximate solutions are usually not available, their development is problem dependent and they may not be accurate enough. Hence, there is a need for methodologies that are capable of generating steady-states for arbitrary unsteady differential models. One attempt in this direction is the selective frequency damping technique, despite being developed for problems with a well defined self-excitation frequency. Another attempt to do so is the physical-time damping technique, but temporal dissipation is proportional to the time step. Since numerical instability can keep this time step too small in many nonlinear problems, this technique may not be able to introduce enough dissipation for the damping of all perturbations in very unstable flows. The present work overcomes this problem by noting that optimal damping is not introduced through maximum temporal dissipation, but minimal gain. The implicit Euler scheme employed in the physical-time damping technique achieves both in the limit of infinite CFL numbers, which usually cannot be imposed due to nonlinear effects. This time marching scheme was modified in order for its minimal gain to occur at smaller CFL numbers. Several test cases confirm the efficacy of this new approach.