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

Abstract

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, Nc; 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 Nc 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.