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Lemma 331 no explicit runge kutta method can have an

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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...
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