Global spectral analysis of three-time level integration schemes: Focusing phenomenon

Here, we explain the phenomenon of focusing using the numerical properties of space-time discretization methods involving second-order Adams-Bashforth (AB2) method for the solution of one-dimensional (1D) convection equation. It has been established that solving 1D convection equation by three-time level method invokes a numerical or spurious mode, apart from the physical mode (as explained in Sengupta et al., [27]). Here, the long elusive problem of focusing (considered as a problem of non-linear numerical aspect), is shown due to a linear mechanism. The focusing is shown for a wave-packet propagating in a non-periodic domain by a three-time level method. Long time integration shows the physical mode to cause focusing, which shows up as spectacular growth of error-packet(s) at discrete location(s), where the dominant wavenumber (k) depends only on the CFL number (Nc), for the space-time discretization method. The length scale of growing error is independent of wavenumber of the input signal. It is also established that focusing is related to numerical absolute instability, for which the numerical group velocity (VgN1) of the physical mode is zero. However, interestingly, when a compact filter is used, the focusing phenomenon is converted from absolute to convective numerical instability. This brings new insight and satisfactory explanation of focusing and its dependence on the choice of numerical methods and use of filter. As a demonstration of the focusing phenomenon for AB2 method, we use it with a well known combined compact differencing scheme to solve Navier-Stokes equation in a square lid driven cavity for a super-critical post-Hopf bifurcation Reynolds number of 10,000 (based on the side of the cavity and the constant lid velocity). Contrary to the well-established solution with polygonal vortices in the literature, here the solution breaks down after a finite time due to focusing.