E y f t y0 y0 0 this problem which is of little

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: is of little interest, can be solved by quadrature to yield Zt y(t) = y0 + f ( )d : 0 We can easily construct high-order approximate methods for this problem by using numerical integration. Thus, for example, the simple left-rectangular rule would lead to Euler's method. The midpoint rule with a step size of h would give us y(h) = y0 + hf (h=2) + O(h3): Thus, by shifting the evaluation point to the center of the interval we obtained a higherorder approximation. Neglecting the local error term and generalizing the method to the interval tn 1 < t tn yields ; yn = yn 1 + hf (tn 1 + h=2): ; ; 2 Runge 21] sought to extend this idea to true di erential equations having the form of (3.1.1). Thus, we might consider yn = yn 1 + hf (tn ; h=2 yn ; 1=2 ; ) as an extension of the simple midpoint rule to (3.1.1). The question of how to de ne the numerical solution yn 1=2 at the center of the interval remains unanswered. A simple possibility that immediately comes to mind is to evaluate it by Euler's method. This gives yn 1=2 = yn 1 + h f (tn 1 yn 1) 2 ; ; ; ; ;...
View Full Document

This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

Ask a homework question - tutors are online