A discrete calculus analysis of the Keller Box scheme and a generalization of the method to arbitrary meshes J.B. Perot * , V. Subramanian University of Massachusetts, Amherst, Mechanical and Industrial Engineering, Amherst, MA 1003, United States Received 16 January 2007; received in revised form 17 April 2007; accepted 20 April 2007 Available online 29 April 2007 Abstract The Keller Box scheme is a face-based method for solving partial differential equations that has numerous attractive mathematical and physical properties. It is shown that these attractive properties collectively follow from the fact that the scheme discretizes partial derivatives exactly and only makes approximations in the algebraic constitutive relations appearing in the PDE. The exact discrete calculus associated with the Keller Box scheme is shown to be fundamentally different from all other mimetic (physics capturing) numerical methods. This suggests that a unique exact discrete calculus does not exist. It also suggests that existing analysis techniques based on concepts in algebraic topology (in particular – the discrete de Rham complex) are unnecessarily narrowly focused since they do not capture the Keller Box scheme. The discrete calculus analysis allows a generalization of the Keller Box scheme to non-simplectic meshes to be constructed. Analysis and tests of the method on the unsteady advection–diffusion equations are presented. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Discrete calculus; Keller Box; Mimetic; Numerical methods 1. Introduction The Keller Box scheme [1] is also sometimes referred to as the Preissman Box scheme [2]. It is a variation of the finite volume approach in which unknowns are stored at control volume faces rather than at the more tra- ditional cell centers. The name alludes to the fact that in space–time, the unknowns sit at the corners of the space–time control volume – which is a box in one space dimension on a stationary mesh. The original devel- opment of the method [1,2] dealt with parabolic initial value problems such as the unsteady heat equation. The method was made better known by Cebeci and Bradshaw [3,4] as a method for the solution of the boundary layer equations. Since that time the approach has been extended to address convection [5,6] and to the Euler and Navier–Stokes compressible equations [7–9]. Some mixed finite element methods [10–12] place some of the degrees of freedom on the element faces (rather than on the vertices). This is a similar idea, however mixed FE 0021-9991/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2007.04.015 * Corresponding author. Tel.: +1 413 545 3925; fax: +1 413 545 1027. E-mail address: perot@ecs.umass.edu (J.B. Perot). Journal of Computational Physics 226 (2007) 494–508 www.elsevier.com/locate/jcp