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

We derive conservative fourth- and sixth-order finite difference approximations for the divergence and gradient operators and a compatible inner product on staggered 1D uniform grids in a bounded domain. The methods combine standard centered difference formulas in the interior with new one-sided finite difference approximations near the boundaries. We derive compatible inner products for these difference methods that are high-order approximations of the continuum inner product. We also investigate defining compatible high-order divergence and gradient finite difference operators that satisfy a discrete integration by parts identity.