Group classification applications for analysis of discrete models of flow in porous media

Abstract

This work is about examples of applications of group classifications for continuous (differential equations) and discrete models of flow in porous media. Difference schemes and discrete dynamical systems are considered as discrete models. Group classifications has been carried out on the basis of continuous groups (Lie groups) of transformations in n-dimensional Euclidean spaces. The example of explicit difference scheme for Buckley-Leverett equation has been demonstrated where the correspondence to one of classes of the obtained earlier classifications of discrete dynamical system has been shown. The example of fractal capillary network on the basis of Sierpinski triangle has been demonstrated for the group classification of discrete dynamical systems. The approach of continuous symmetry absence or presence proof has been demonstrated using classifications of discrete dynamical systems. Difference schemes with continuous symmetry groups for partial differential equations of parabolic type, which correspond to the equation of gas flow in porous media and Rapoport-Leas equation (generalized Buckley-Leverett equation), are considered. Examples of difference schemes for these cases are given and related to classes of the obtained earlier classifications. Applications of continuous symmetry groups for numerical solutions generations are demonstrated for these cases.