Abstract
Discrete maximum principles for linear equation systems are discussed. A novel maximum principle for linear difference schemes is established. Sufficient conditions for an arbitrary linear scheme to satisfy the maximum principle are provided. Easily verifiable sufficient as well as necessary and sufficient conditions of nonsingularity for a diagonally dominant matrix, be it reducible or irreducible, are derived. Necessary and sufficient conditions for the validity of the maximum principle for explicit difference schemes are developed. The notion of submonotonicity for linear difference schemes as well as the notions of linear monotonicity and linear submonotonicity for nonlinear difference schemes are introduced and associated criteria developed. The developed approaches are demonstrated by examples of known linear and nonlinear difference schemes associated, in general, with the numerical analysis of systems of partial differential equations.
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