Causes of instabilities in numerial integration techniques

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Abstract
This paper presents the causes of instabilities which arise during the numerical solution of ordinary differential equations. Using the numerical integration routines presently available, one actually approximates the differential equation by a difference equation. If the difference equation is of higher order than the original differential equation, the approximate solution contains extraneous solutions which are not at all related to the true solution. It is the behavior of these extraneous solutions that one is concerned with in a stability analysis.