Symmetry-preserving discrete schemes for some heat transfer equations
  Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterparts. 
Margarita Bakirova, Vladimir Dorodnitsyn and Roman Kozlov