INVARIANT VARIATIONAL-DIFFERENCE SCHEMES AND CONSERVATION-LAWS

A study is presented of the connection between the conservation laws of differential-difference schemes (differential with respect to time, difference with respect to space) in the mechanics of continuous media, in Lagrangian variables, on the one hand, and transformation groups, on the other. Noether's Theorem is generalized to a class of differential-difference schemes that possess an equivalent variational formulation. It is shown that a necessary and sufficient condition for this class to have a conservation law is invariance of the extremal values of a spatially discrete variational functional. Schemes for gas dynamics and incompressible liquids are considered as examples. Examples of computations are given.