Global spectral analysis of the Lax–Wendroff-central difference scheme applied to Convection–Diffusion equation

Abstract

In this work, the Global Spectral Analysis (GSA) is applied to the convective Lax–Wendroff based discretization of linear convection–diffusion problem in both 1D and 2D. Contrary to standard numerical analysis approaches, two important physical processes (convection and diffusion) are treated together, thus making GSA a function of multiple non-dimensional numerical parameters, namely, the non-dimensional wavenumber (kh), the Courant–Friedrich–Lewy number (Nc) and the Peclet number (Pe). All the three quantities impact the stability of the numerical scheme by affecting both numerical amplification factor as well as numerical diffusion. Likewise, numerical phase speed and numerical group velocity all become expressions of all three non-dimensional parameters. Leveraging the GSA, an accurate map (property charts) of acceptable range of parameter space is evidenced to obtain the spatio-temporal numerical solution that is stable as well as Dispersion Relation Preserving (DRP). The numerical property charts are shown to be useful to calibrate numerical solutions of both 1D and the 2D convection–diffusion equations on uniform meshes. Finally, as demonstrated while solving the Navier–Stokes equation for a Taylor–Green vortex problem, property charts allow to explain numerical behaviors often observed with real complex flow problems.