For purposes of numerical integration, differential equations are often modeled by finite-differences. However, these models raise a number of questions related to how the finite-difference schemes are to be constructed, the magnitude of the local truncation errors, the existence and elimination of numerical instabilities, etc. We prove a theorem which states that to each ordinary differential equation there corresponds an “exact” finite-difference scheme, i.e., the local truncation error is zero. This means that on the computational grid or lattice, the solution to the difference equation is exactly equal to the solution to the differential equation. We use the theorem to show, by means of a number of explicit examples, that the usual rules for constructing finite-difference schemes are “wrong.” Several modeling principles are presented, as well as their application to partial differential equations.
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