  The global spectral analysis (GSA) of numerical methods ensures that the
dispersion relation preserving (DRP) property is calibrated in addition to
ensuring numerical stability, as advocated in the von Neumann analysis. The DRP
nature plays a major role where spatio-temporal dependence in the governing
equation and boundary conditions has to be retained, such as in direct
numerical simulations (DNS) and large eddy simulations (LES) of fluid flow
transition. Using the concept of GSA, methods based on the Lax-Wendroff
approach for temporal integration are calibrated using a high accuracy, sixth
order non-uniform compact scheme, developed in "Hybrid sixth order spatial
discretization scheme for non-uniform Cartesian grids - Sharma et al. Comput.
Fluids, 157, 208-231 (2017)." The model equation used for this analysis is the
one-dimensional (1D) convection-diffusion equation (CDE) which provides a
unique state for the Lax-Wendroff method, results of which will have direct
consequences for the solution of Navier-Stokes equations. Furthermore, the
specific choice of the governing equation enables a direct assessment of the
performance of numerical methods for solving fluid flows due to its one-to-one
correspondence with the Navier-Stokes equation as established in "Effects of
numerical anti-diffusion in closed unsteady flows governed by two-dimensional
Navier-Stokes equation - Suman et al. Comput. Fluids, 201, 104479 (2020)". The
limiting case of the non-uniform compact scheme, which is a uniform grid, is
considered. This is also investigated using GSA, and potential differences for
the non-uniformity of grid are compared. Finally, further use of this newly
developed Lax-Wendroff method for the non-uniformity of grid is quantified for
its application in DNS and LES.
