Temporal dynamics of the march-in-time toward the steady solution of advection–diffusion systems

ABSTRACT

This paper addresses the temporal dynamics of the time marching when this approach is used for determining the steady solution of advection–diffusion problems. The time-step size, $\delta t$, is the main parameter and $\mathbb{N}\left(\delta t\right)$ is the number of time steps which is needed to reach the steady solution, from a given initial state and according to a pre-defined time-convergence criterion. The measure of $\mathbb{N}\left(\delta t\right)$ obtains by proceeding to march in time with a large set of $\delta t$ values. Four distinct advection–diffusion problems are considered. They are time discretized by a usual second-order Finite-Difference scheme, and their spatial behaviour is computed by using two different spectral solvers. The space-grid parameter is N, the spectral cut-off for any space direction. A universal $\mathbb{N}\left(\delta t\right)$ behaviour is observed, giving rise to V-shaped plots when the Courant-Friedrich-Levy instability imposes its control to the march-in-time. Moreover, one of the solvers leads to U-shaped plots for the incompressible-flow configurations in weakly nonlinear regimes. This is the temporal consequence of the time-splitting method that this solver has adopted for uncoupling the velocity from the pressure.

KEYWORDS:

• Temporal dynamics
• advection–diffusion
• optimal $\delta t$
• CFL limit optimal $N_{\mathrm{opt}}\left(\delta t_{\mathrm{opt}}\right)$

2010 AMS SUBJECT CLASSIFICATIONS:

• 35J05
• 65N12
• 65N35

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

LZC would like to acknowledge the support from National Natural Science Foundation of China through Grant [11671166] and [U1530401], China Postdoctoral Science Foundation through Grant [2015M580038]. The second author is particularly grateful to Professor Li-Shi Luo for his kind invitation to visit CSRC.