ORDER OF ACCURACY OF QUICK AND RELATED CONVECTION-DIFFUSION SCHEMES

This paper explains significant differences in trunication error between finite-difference and finite-volume convection-diffusion schemes. Specifically, the order of accuracy of the QUICK scheme for steady-state convection and diffusion is discussed in detail. Other related convection-diffusion schemes are also considered. The original one-dimensional QUICK scheme written in terms of nodal-point values of the convected variable (with a 1/8-factor multiplying the ''curuature'' term) is indeed a third-order representation of the finite-volume formulation of the convection operator average across the control volume, written naturally in flux-difference form. An alternative single-point upwind difference scheme (SPUDS) using node values (with a 1/6-factor) is a third-order representation of the finite-difference single-point formulation; this can be written in a pseudo-flux-difference form. These are both third-order convection schemes; however, the QUICK finite-volume convection operator is 33% more accurate than the single-point implementation of SPUDS. Another finite-volume scheme, writing convective fluxes in terms of cell-average values, requires a 1/6-factor for third-order accuracy. For completeness, one can also write a single-point formulation of the convective derivative in terms of cell averages and then express this in pseudo-flux-difference form; for third-order accuracy, this requires a curvature factor of 5/24. Diffusion operators are also considered in both finite-difference and finite-volume formulations. Finite-volume formulations are found to be significantly more accurate. For example classical second-order central differencing for the second derivative is exactly twice as accurate in a finite-volume formulation as it is in a finite-difference formulation.