On the relation between elementary partial difference equations and partial differential equations

The nonstandard stroboscopy method links discrete-time ordinary difference equations of first-order and continuous-time, ordinary differential equations of first order. We extend this method to the second order, and also to an elementary, yet general class of partial difference/differential equations, both of first and second order. We thus obtain straightforward discretizations and continuizations, even avoiding change of variables. In fact, we create intermediary objects: partial difference equations with S-continuous solutions, which have both discrete and continuous properties. (C) 1998 Elsevier Science B.V. All rights reserved.