  We present the multiplier method of constructing conservative finite
difference schemes for ordinary and partial differential equations. Given a
system of differential equations possessing conservation laws, our approach is
based on discretizing conservation law multipliers and their associated density
and flux functions. We show that the proposed discretization is consistent for
any order of accuracy when the discrete multiplier has a multiplicative
inverse. Moreover, we show that by construction, discrete densities can be
exactly conserved. In particular, the multiplier method does not require the
system to possess a Hamiltonian or variational structure. Examples, including
dissipative problems, are given to illustrate the method. In the case when the
inverse of the discrete multiplier becomes singular, consistency of the method
is also established for scalar ODEs provided the discrete multiplier and
density are zero-compatible.
