Error growth and phase lag analysis for high Courant numbers

摘要

The type of numerical scheme used in numerical computation of fluid flow largely affects the accuracy of the solution. In order to obtain accurate numerical solution in a transient analysis, implicit schemes are known to be unconditionally stable for linear systems while non-linearity in advection systems often limits the stability. Nevertheless, the implicit schemes for non-linear systems allow larger Courant numbers than explicit schemes. However, these assertions have been addressed based on a simple numerical error growth analysis which does not account for dispersion and phase lag errors even in linear systems. Thus, motivated by the correct error dynamics given in Sengupta et al., (2007), it has been shown here that stable and accurate numerical solutions are possible for higher Courant numbers while their low values do not always lead to stable and accurate solutions. Here, higher Courant numbers can be achieved by finer grid spacing, keeping the time step same. Time integration is performed by implicit scheme, for which numerical stability is guaranteed and thus a higher Courant number implies higher resolution of the solution. This has been proven through a test case of the propagation of acoustic wave in water by considering the latter to be a compressible medium by solving unsteady Navier–Stokes equations. This observation is elucidated by evaluating the error growth with first order in time and second order in space schemes with the latter represented by fully implicit form at t = (n + 1)Δt. The results show that for high Courant numbers, the value of error growth is very close to unity and confirms to the needed neutral stability of the obtained numerical solution. Furthermore, the value of phase error nearly vanishes and reveals that numerical waves travel with nearly the same speed as physical waves.