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# Lemma 321 all p stage explicit runge kutta methods of

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Unformatted text preview: bsolute-stability region is determined as ; 1 + z = ei which can easily be recognized as the familiar unit circle centered at z = ;1 + 0i. For real values of z the intervals of absolute stability for methods with p = s 4 are shown in Table 3.2.3. Absolute stability regions for complex values of z are illustrated for the same methods in Figure 3.2.1. Methods are stable within the closed regions shown. The regions of absolute stability grow with increasing p. When p = 3 4, they also extend slightly into the right half of the complex z-plane. Problems 1. Instead of solving the IVP (3.1.1), many software systems treat an autonomous ODE y = f (y). Non-autonomous ODEs can be written as autonomous systems 0 16 Order, p Interval of Absolute Stability 1 (-2,0) 2 (-2,0) 3 (-2.51,0) 4 (-2.78,0) Table 3.2.3: Interval of absolute stability for p-stage explicit Runge-Kutta methods of order p = 1 2 3 4. 3 2 Im(z) 1 0 −1 −2 −3 −5 −4 −3 −2 Re(z) −1 0 Figure 3.2.1: Region of absolute stability for p-stage explicit R...
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