Runge_Kutta.ppt

5579 4 10 81 2 1200 8 4 12 1 f t f k o 09

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5579 . 4 10 81 1200 10 2067 . 2 1200 , 0 , 8 4 12 0 1 f t f k o 017595 . 0 10 81 09 . 106 10 2067 . 2 09 . 106 , 240 240 5579 . 4 1200 , 240 0 , 8 4 12 1 0 0 2 f f h k h t f k K h k k 16 . 655 240 2702 . 2 1200 240 017595 . 0 2 1 5579 . 4 2 1 1200 2 1 2 1 2 1 0 1
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Solution Cont Step 2: K h t t i 16 . 655 , 240 240 0 , 1 1 0 1 38869 . 0 10 81 16 . 655 10 2067 . 2 16 . 655 , 240 , 8 4 12 1 1 1 f t f k 20206 . 0 10 81 87 . 561 10 2067 . 2 87 . 561 , 480 240 38869 . 0 16 . 655 , 240 240 , 8 4 12 1 1 1 2 f f h k h t f k K h k k 27 . 584 240 29538 . 0 16 . 655 240 20206 . 0 2 1 38869 . 0 2 1 16 . 655 2 1 2 1 2 1 1 2
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Solution Cont The exact solution of the ordinary differential equation is given by the solution of a non-linear equation as 9282 . 2 10 22067 . 0 0033333 . 0 tan 8519 . 1 300 300 ln 92593 . 0 3 1 t The solution to this nonlinear equation at t=480 seconds is K 57 . 647 ) 480 (
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Comparison with exact results Figure 2. Heun’s method results for different step sizes
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Effect of step size Table 1. Temperature at 480 seconds as a function of step size, h Step size, h (480) E t t |% 480 240 120 60 30 −393.87 584.27 651.35 649.91 648.21 1041.4 63.304 −3.7762 −2.3406 −0.63219 160.82 9.7756 0.58313 0.36145 0.097625 K 57 . 647 ) 480 ( (exact)
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Effects of step size on Heun’s Method Figure 3. Effect of step size in Heun’s method
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Step size, h (480) Euler Heun Midpoint Ralston 480 240 120 60 30 −987.84 110.32 546.77 614.97 632.77 −393.87 584.27 651.35 649.91 648.21 1208.4 976.87 690.20 654.85 649.02 449.78 690.01 667.71 652.25 648.61 Comparison of Euler and Runge-Kutta 2 nd Order Methods Table 2 . Comparison of Euler and the Runge-Kutta methods K 57 . 647 ) 480 ( (exact)
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  • Fall '19
  • Trigraph, Yi, Partial differential equation, y 1.3e  x

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