Lemma 331 no explicit runge kutta method can have an

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 bounded. Hence, once again, we turn to implicit methods as a means of enlarging the region of absolute stability. Necessary order conditions for s-stage implicit Runge-Kutta methods are given by (3.2.5c, 3.2.7) (with su cient conditions given in Hairer et al. 16], Section II.2). A condition on the maximum possible order follows. Theorem 3.3.1. The maximum order of an implicit s-stage Runge-Kutta method is 2s. 18 Proof. cf. Butcher 7]. The derivations of implicit Runge-Kutta methods follow those for explicit methods. We'll derive the simplest method and then give a few more examples. Example 3.3.1. Consider the implicit 1-stage method obtained from (3.2.3) with s = 1 as yn = yn 1 + hb1 f (tn 1 + c1h Y1) (3.3.1a) Y1 = yn 1 + ha11 f (tn 1 + c1h Y1): (3.3.1b) ; ; ; ; To determine the coe cients c1, b1 , and a11 , we substitute the exact ODE solution into (3.3.1a,b) and expand (3.3.1a) in a Taylor's series y(tn) = y(tn 1) + hb1 f + c1hft + fy (Y1 ; y(tn 1)) + O(h2)] ; ; where f := f (tn 1 y(tn 1)), etc. Expanding (3.3.1b) in a Taylor's se...
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