We present the multiplier method of constructing conservative finitedifference schemes for ordinary and partial differential equations. Given asystem of differential equations possessing conservation laws, our approach isbased on discretizing conservation law multipliers and their associated densityand flux functions. We show that the proposed discretization is consistent forany order of accuracy when the discrete multiplier has a multiplicativeinverse. Moreover, we show that by construction, discrete densities can beexactly conserved. In particular, the multiplier method does not require thesystem to possess a Hamiltonian or variational structure. Examples, includingdissipative problems, are given to illustrate the method. In the case when theinverse of the discrete multiplier becomes singular, consistency of the methodis also established for scalar ODEs provided the discrete multiplier anddensity are zero-compatible.
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