Fourth- and sixth-order conservative finite difference approximations of the divergence and gradient

We derive conservative fourth- and sixth-order finite difference approximat ions for the divergence and gradient operators and a compatible inner produ ct on staggered 1D uniform grids in a bounded domain. The methods combine s tandard centered difference formulas in the interior with new one-sided fin ite difference approximations near the boundaries, We derive compatible inn er products for these difference methods that are high-order approximations of the continuum inner product. We also investigate defining compatible hi gh-order divergence and gradient finite difference operators that satisfy a discrete integration by parts identity. (C) 2001 IMACS. Published by Elsev ier Science B.V. All rights reserved.