Consistency conditions for the influence graphs generalized finite difference method

Abstract

The modeling of physical phenomena is most often conveyed through partial differential equations governing continuous fields. Computational approaches are used afterwards in order to approximate the solutions to these equations. In doing so, a discretized version of the initial model must be set up using one of the numerous available methods. This paper illustrates how influence graphs (weighted digraphs with geometrical attributes) may behave in the same way as differential operators in so far as a set of appropriate conditions of consistency are satisfied. These conditions are written explicitly for the most often used cases, including differential boundary operators. On the one hand, a consistent neighborhood may be regarded as a generalization of a classical finite difference scheme. On the other hand, a consistent influence graph is a discrete object modeling a physical behavior as well as its continuous counterpart. Numerical experimentations address firstly the second order diffusion model and lastly the fourth order Kirchhoff theory of plate deflection.