Influence graphs and the generalized finite difference method

This paper introduces directed graphs on which the evolution of a physical quantity depends only on local neighborhoods. These graphs are then used to model transfer phenomena occurring under a convective mode or a diffusive mode. The conditions under which the state associated to such graphs approaches the solution of a diffusion-convection partial differential model are established. An algorithm permitting to determine consistent neighborhoods is described and recognized as a generalization of the finite difference method. Examples of application are presented in order to illustrate the practical applicability of these concepts.