High Order Finite Di®erencing Schemes and Their Accuracy for CFD

This paper investigates di®erent high order ¯nite di®erence schemes and their accuracy for Burgers equation and Navier-Stokes equations. On a coarse grid, theoretical and numerical analysis indicate that a higher order di®erence scheme does not necessarily obtain more accurate solutions than a lower order scheme in the regions of high gradient variation (2nd order derivative). On a coarse grid, the numerical experiments indicate that the 3rd order biased upwind di®erencing for the inviscid °uxes with 2nd order central di®erencing for viscous term outperforms all other schemes with the accuracy order as high as 7th order. For the linear Burgers equation, both the analytical and numerical results indicate that a higher order scheme will be more accurate than a lower order scheme only when Re¢x · 1. For the nonlinear Burgers equation, the numerical experiments also show the same conclusion. A numerical solution of laminar wall boundary layer from Navier-Stokes equations shows the same observation that a lower order scheme is more accurate on a coarse grid. The conception that a high order scheme can reduce mesh size unconditionally may be incorrect. Since the discrepancy occurs mostly in the region with high gradient variation, the re¯ned mesh should not only be near the wall surface where the gradient is high, but also near the edge of the boundary layer where the gradient variation is high. The mesh spacing criterion for Navier-Stokes equation that a higher order scheme will be more accurate than a lower order scheme is not obtained yet. Using the criterion of the linear Burgers equation for Navier-Stokes equations may be excessive and preventively expensive. To resolve a laminar wall boundary layer accurately, the recommended grid Reynolds number is to be below 25 across the boundary layer. However, this is the criterion from the numerical experiments for all the ¯nite di®erencing schemes, not for the higher order schemes to have better accuracy.