A discrete calculus analysis of the Keller Box scheme and a generalization of the method to arbitrary meshes

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.