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

This paper reports a comparative study on the stability limits of nine fini te 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-ord er upwind; (v) second-order central differences; and (vi) first-order upwin d. These schemes were used together with Runge-Kutta temporal discretizatio ns up to order six. The remaining schemes are the (vii) Adams-Bashforth cen tral differences, (viii) the Quickest and (ix) the Leapfrog central differe nces. In addition, the dispersive and dissipative characteristics of the sc hemes 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, toget her with central schemes, show good conditional stability limits and good d ispersive and dissipative spectral resolution. Overall the fourth-order com pact is the recommended scheme. Copyright (C) 2001 John Wiley & Sons, Ltd.