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Unformatted text preview: Practical Session 7 — Solutions Exercises 1. (a) The ODE to be solved is dy/dx = 4 / (1 + x 2 ) , integrated from x = 0 to x = 1 with the initial condition y (0) = 0 . The numerical value of y (1) is then the estimate of the integral. (b) The modifications are to change the upper limit of integration to 1.0 and the RHS of the ODE in derivs() to dydx = 4.0/(1.0 + xin*xin); . The result is in error by about 0.03%. 2. (a) This only requires replacing rk2() by euler() . The result is shown in Figure 1 — the amplitudes of the numerical solutions increase with x , rather than remaining constant.-2-1.5-1-0.5 0.5 1 1.5 2 2.5 2 4 6 8 10 12 14 y, z x "harmonic_e_100.txt" using 1:2 "harmonic_e_100.txt" using 1:3 sin(x) cos(x) Figure 1: Result for 2. (a). (b) The phase space diagram is shown in Figure 2. The solution spirals out rather than de- scribing a circle with unit radius. (c) The results are shown in Figure 3. For increased numbers of steps the solution is more accurate, but still exhibits a growing amplitude of oscillations with increasing x . For 1000 steps the result is becoming accurate, but the problem is still there. (d) Figure 4 shows the results. The amplitudes increase linearly on a log-linear plot, indicating that the amplitudes grow exponentially, which is characteristic of instability. (e) Euler’s method is unstable when applied to this problem. By comparison, Runge-Kutta is quite stable. 3. (1002) (a) The ODE is dy dx = 2 π h 1- sin 2 π 4 sin 2 x i- 1 / 2 , and this needs to be integrated from x = 0 to x = π 2 with the initial condition y (0) = 0 , in which case y ( π/ 2) is the estimate of the integral....
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This note was uploaded on 09/29/2009 for the course COSC 1002 taught by Professor Wheatland during the Three '09 term at University of Sydney.
- Three '09