Effects of numerical anti-diffusion in closed unsteady flows governed by two-dimensional NavierStokes equation

Numerical methods producing acceptable results for a long time abruptly blow up, without providing any indication of localized onset of sudden numerical instability. This has been identified as focusing problem in literature. It is noted that the scale selection of error does not depend on the relevant excited physical space-time scales. While this has been encountered in weather prediction studies, it is not widely reported from the solution of Navier-Stokes equation (NSE). Recently, in “Focusing phenomenon in numerical solution of two-dimensional Navier-Stokes equation, In: Pirozzoli S., Sengupta T. (eds) High-Performance Computing of Big Data for Turbulence and Combustion, CISM International Centre for Mechanical Sciences (Courses and Lectures), vol 592. Springer, Cham (2019)”, focusing was demonstrated for a steady fluid flow and its mechanism was identified from global spectral analysis (GSA) of 2D convection diffusion equation (CDE). Focusing was shown to be due to the anti-diffusion caused by the discretization of diffusion term for the chosen numerical scheme. The present work consolidates the one-to-one correspondence between numerical anti-diffusion of 2D CDE and focusing for unsteady flows by solving flow inside a 2D lid driven cavity (LDC) for the Reynolds number of 10,000. We also present a method to remove numerical anti-diffusion using multi-dimensional filters. Detailed analysis of space-time discretization with filters is also provided to explain the cure of focusing

Global spectral analysis for convection-diffusion-reaction equation in one and two-dimensions: Effects of numerical anti-diffusion and dispersion

Convection-diffusion-reaction (CDR) equation plays a central role in many disciplines of engineering, science and finances. As a consequence, importance of analysis of numerical methods for the accurate solution of CDR equation has motivated the present research. We have used the global spectral analysis to characterize all the three important physical processes in terms of the non-dimensional numerical parameters, namely, the non-dimensional wavenumber (kh); Courant-Friedrich-Lewy (CFL) number, ; the Peclet number (Pe) and the Damkohler number (Da). For the purpose of illustration, we have focused on two space-time discretization schemes known for accuracy and robustness. The basic properties relate to numerical issues arising for the numerical amplification factor, numerical diffusion coefficient, numerical phase speed and numerical group velocity. With the help of model one-dimensional (1D) and two-dimensional (2D) CDR equations, we have reported the numerical property charts for the cases: (i) When all the processes of convection, diffusion and reaction are of same order, with critical numerical behaviour enforcing low values of Da for the 1D CDR equation studied here. (ii) The 2D CDR equation considered is diffusion-reaction dominated, and as a consequence, this enforces Da to be larger. We have thoroughly analyzed these cases to identify the essential roles of anti-diffusion on the critical and Pe values, which in turn decides admissible space and time steps to be used with the discretization schemes. The property charts have been used to calibrate the analysis with two model equations, one of which has an exact solution for a 1D CDR equation, and the second case for the 2D CDR equation has numerical solution available in the literature. These cases help to identify the importance of such analysis in explaining the utility of the choice one can exercise in fixing the numerical parameters. This also identifies and explains some hitherto unknown numerical problems for CDR equation and their alleviation techniques