Improved Eulers Method Improved Eulers Method Algorithm Example Improved Eulers

Improved eulers method improved eulers method

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Improved Euler’s Method Improved Euler’s Method - Algorithm Example Improved Euler’s Method Error Order of Error Numerical Example 1 Numerical Example: Consider the IVP dy dt = 2 e - 0 . 1 t - sin( y ) , y (0) = 3 , which has no exact solution, so must solve numerically Solve this problem with Euler’s method and Improved Euler’s method Show differences with different stepsizes for t [0 , 5] Show the order of convergence by halving the stepsize twice Graph the solution and compare to solution from ode23 in MatLab, closely approximating the exact solution Joseph M. Mahaffy, h [email protected] i Lecture Notes – Numerical Methods for Differe — (35/39)
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Introduction Euler’s Method Improved Euler’s Method Improved Euler’s Method - Algorithm Example Improved Euler’s Method Error Order of Error Numerical Example 2 Numerical Solution for dy dt = 2 e - 0 . 1 t - sin( y ) , y (0) = 3 Used MatLab’s ode45 to obtain an accurate numerical solution to compare Euler’s method and Improved Euler’s method with stepsizes h = 0 . 2, h = 0 . 1, and h = 0 . 05 “Actual” Euler Im Eul Euler Im Eul Euler Im Eul t n h = 0 . 2 h = 0 . 2 h = 0 . 1 h = 0 . 1 h = 0 . 05 h = 0 . 05 0 3 3 3 3 3 3 3 1 5.5415 5.4455 5.5206 5.4981 5.5361 5.5209 5.5401 2 7.1032 7.1718 7.0881 7.1368 7.0995 7.1199 7.1023 3 7.753 7.836 7.743 7.7939 7.7505 7.7734 7.7524 4 8.1774 8.2818 8.167 8.2288 8.1748 8.2029 8.1768 5 8.5941 8.7558 8.5774 8.6737 8.5899 8.6336 8.5931 1 . 88% - 0 . 194% 0 . 926% - 0 . 0489% 0 . 460% - 0 . 0116% Last row shows percent error between the different approximations and the accurate solution Joseph M. Mahaffy, h [email protected] i Lecture Notes – Numerical Methods for Differe — (36/39)
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Introduction Euler’s Method Improved Euler’s Method Improved Euler’s Method - Algorithm Example Improved Euler’s Method Error Order of Error Numerical Example 3 Error of Numerical Solutions Observe that the Improved Euler’s method with stepsize h = 0 . 2 is more accurate at t = 5 than Euler’s method with stepsize h = 0 . 05 With Euler’s method the error cuts in half with halving of the stepsize With the Improved Euler’s method the errors cuts in quarter with halving of the stepsize Joseph M. Mahaffy, h [email protected] i Lecture Notes – Numerical Methods for Differe — (37/39)
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Introduction Euler’s Method Improved Euler’s Method Improved Euler’s Method - Algorithm Example Improved Euler’s Method Error Order of Error Numerical Example 4 Graph of Solution: Actual, Euler’s and Improved Euler’s methods with h = 0 . 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 5 10 15 20 25 30 t y ( t ) dy/dt = y + t Actual Solution Euler Solution Improved Euler The Improved Euler’s solution is very close to the actual solution Joseph M. Mahaffy, h [email protected] i Lecture Notes – Numerical Methods for Differe — (38/39)
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Introduction Euler’s Method Improved Euler’s Method Improved Euler’s Method - Algorithm Example Improved Euler’s Method Error Order of Error Order of Error Error of Numerical Solutions Order of Error without good “Actual solution” Simulate system with stepsizes h , h/ 2, and h/ 4 and define these simulates as y 1 n , y 2 n , and y 3 n , respectively Compute the ratio (from Cauchy sequence) R = | y 3 n - y 2 n | | y 2 n - y 1 n | If the numerical method is order m , then this ratio is approximately 1 2 m Above example at t = 5 has R = 0 . 488 for Euler’s method and R = 0 . 256 for Improved Euler’s method Allows user to determine how much error numerical routine is generating Joseph M. Mahaffy, h [email protected] i Lecture Notes – Numerical Methods for Differe — (39/39)
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