Fourier analysis of several finite difference schemes for the one-dimensional unsteady convection-diffusion equation

摘要

This paper reports a comparative study on the stability limits of nine finite difference schemes to discretize the one-dimensional unsteady convection-diffusion equation. The tested schemes are: (i) fourth-order compact; (ii) fifth-order upwind; (iii) fourth-order central differences; (iv) third-order upwind; (v) second-order central differences; and (vi) first-order upwind. These schemes were used together with Runge-Kutta temporal discretizations up to order six. The remaining schemes are the (vii) Adams-Bashforth central differences, (viii) the Quickest and (ix) the Leapfrog central differences. In addition, the dispersive and dissipative characteristics of the schemes were compared with the exact solution for the pure advection equation, or simple first or second derivatives, and numerical experiments confirm the Fourier analysis. The results show that fourth-order Runge-Kutta, together with central schemes, show good conditional stability limits and good dispersive and dissipative spectral resolution. Overall the fourth-order compact is the recommended scheme.