A symbolic manipulation system such as maple or

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Unformatted text preview: for the most part, we will not need to present detailed derivations of higher-order methods. There are three one-parameter families of three-stage, third-order explicit RungeKutta methods 3, 16]. However, the most popular explicit methods are of order four. Their tableau has the general form 00000 c2 a21 0 0 0 c3 a31 a32 0 0 c4 a41 a42 a43 0 b1 b2 b3 b4 The Taylor's series produce eleven equations for the thirteen nonzero parameters listed above. The classical Runge-Kutta method has the following form: yn = yn 1 + h (k1 + 2k2 + 2k3 + k4) 6 (3.2.4a) k1 = f (tn 1 yn 1) (3.2.4b) k2 = f (tn 1 + h=2 yn 1 + hk1=2) (3.2.4c) ; where ; ; ; ; 9 Order, k 1234567 8 9 Min. Fn. Evals. 1 2 3 4 6 7 9 11 11 Table 3.2.1: Minimum number of function evaluations for explicit Runge-Kutta methods of various orders. k3 = f (tn 1 + h=2 yn 1 + hk2=2) (3.2.4d) k4 = f (tn 1 + h yn 1 + hk3): (3.2.4e) ; ; ; ; Some observations about this method follow: 1. The local error of (3.2.4) is O(h5). In order to get an a priori estimate of the local...
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