Applied Numerical Mathematics 37 (2001) 171–187 Fourth- and sixth-order conservative finite difference approximations of the divergence and gradient ✩ J.E. Castillo a , J.M. Hyman b , M. Shashkov b , S. Steinberg c a Department of Mathematics, San Diego State University, San Diego, CA 92182-7720, USA b Theoretical Division, Mail Stop B284, Los Alamos National Laboratory, Los Alamos, NM 87545, USA c Department of Mathematics, University of New Mexico, Albuquerque, NM 87131, USA Received 30 November 1999; accepted 7 August 2000 Abstract 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.  2001 IMACS. Published by Elsevier Science B.V. All rights reserved. 1. Introduction We are developing a discrete analog of vector and tensor calculus that can be used to accurately approximate continuum models for a wide range of physical processes on logically rectangular, nonorthogonal, nonsmooth grids. These finite difference methods (FDMs) preserve fundamental properties of the original continuum differential operators and allow the discrete approximations of partial differential equations (PDEs) to mimic critical properties, including conservation laws and symmetries, in the solution of the underlying physical problem. The discrete analogs of div, grad, and curl satisfy the identities and theorems of vector and tensor calculus and provide new reliable algorithms for a wide class of PDEs [6–9]. This approach has been used to construct high-quality mimetic finite-difference approximations for the divergence, gradient [16,17], and curl [11]. In [10] this new methodology has been applied to Maxwell’s first-order curl equations. We have created higher-order approximations (see [2,3, 5]) in 1D and 2D on smooth curvilinear grids that satisfy a summation by parts identity for the particular case of periodic boundary conditions. ✩ This research was supported by the Department of Energy under contracts W-7405-ENG-36, the Applied Mathematical Sciences Program KC-07-01-01, and the National Science Foundation Grant CCR-9531828. 0168-9274/01/$ – see front matter  2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII:S0168-9274(00)00033-7