Stability of numerical solution of systems of ordinary differential equations Public Deposited

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  • The background for this paper is the use of quadrature formulas for the solution of ordinary differential equations. If we know the values of the dependent variable for which we are solving, and its derivative, at several equally spaced points, i.e., at values of the independent variable separated by equal intervals, we may use a quadrature formula to integrate the values of the derivative, so that we may obtain an approximate value of the dependent variable at the next point. The differential equation is then used to evaluate the derivative at the new point. This procedure is then repeated to evaluate the dependent variable and its derivative at point after point. The accuracy of this method is limited by the accuracy of the quadrature formula used. In order to improve the accuracy of the solution one may use an open-type quadrature formula to "predict" the value of the dependent variable at the next point, then calculate the derivative, and now use a more accurate closed-type formula to "correct" the value of the dependent variable. This procedure is the basis of "Milne's method". It has been shown that an error introduced at a step propagates itself approximately according to a linear combination of the solutions of a linear difference equation associated with the corrector . The solutions of this difference equation consist of an approximation to the solution of the differential equation and in some cases one or more extraneous solutions. If one or more of the latter increases as the process is repeated from step to step, the method is called instable. Remedies for instability include periodic use of special quadrature formulas called "stabilizers". This has been treated in the case of fifth-order formulas by Milne and Reynolds. In this paper the idea is extended to formulas of seventh order.
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