Improved Eulers Method Error Order of Error Example Improved Eulers Method 2

Improved eulers method error order of error example

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Improved Euler’s Method Error Order of Error Example: Improved Euler’s Method 2 Solution: Let y 0 = 3, the Euler’s formula is y n +1 = y n + h ( y n + t n ) = y n + 0 . 1( y n + t n ) The Improved Euler’s formula is ye n = y n + h ( y n + t n ) = y n + 0 . 1( y n + t n ) with y n +1 = y n + h 2 (( y n + t n ) + ( ye n + t n + h )) y n +1 = y n + 0 . 05 ( y n + ye n + 2 t n + 0 . 1) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Numerical Methods for Differe — (30/39)
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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 Example: Improved Euler’s Method 3 Solution: Below is a table of the numerical computations t Euler’s Method Improved Euler Actual 0 y 0 = 3 y 0 = 3 y (0) = 3 0 . 1 y 1 = 3 . 3 y 1 = 3 . 32 y (0 . 1) = 3 . 3207 0 . 2 y 2 = 3 . 64 y 2 = 3 . 6841 y (0 . 2) = 3 . 6856 0 . 3 y 3 = 4 . 024 y 3 = 4 . 0969 y (0 . 3) = 4 . 0994 0 . 4 y 4 = 4 . 4564 y 4 = 4 . 5636 y (0 . 4) = 4 . 5673 0 . 5 y 5 = 4 . 9420 y 5 = 5 . 0898 y (0 . 5) = 5 . 0949 0 . 6 y 6 = 5 . 4862 y 6 = 5 . 6817 y (0 . 6) = 5 . 6885 0 . 7 y 7 = 6 . 0949 y 7 = 6 . 3463 y (0 . 7) = 6 . 3550 0 . 8 y 8 = 6 . 7744 y 8 = 7 . 0912 y (0 . 8) = 7 . 1022 0 . 9 y 9 = 7 . 5318 y 9 = 7 . 9247 y (0 . 9) = 7 . 9384 1 y 10 = 8 . 3750 y 10 = 8 . 8563 y (1) = 8 . 8731 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Numerical Methods for Differe — (31/39)
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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 Example: Improved Euler’s Method 4 Graph of Solution: Actual, Euler’s and Improved Euler’s 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 — (32/39)
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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 Example: Improved Euler’s Method 5 Solution: Comparison of the numerical simulations It is very clear that the Improved Euler’s method does a substantially better job of tracking the actual solution The Improved Euler’s method requires only one additional function, f ( t, y ), evaluation for this improved accuracy At t = 1, the Euler’s method has a - 5 . 6% error from the actual solution At t = 1, the Improved Euler’s method has a - 0 . 19% error from the actual solution Joseph M. Mahaffy, h [email protected] i Lecture Notes – Numerical Methods for Differe — (33/39)
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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 Improved Euler’s Method Error Improved Euler’s Method Error Showed earlier that Euler’s method had a local truncation error of O ( h 2 ) with global error being O ( h ) Similar Taylor expansions (in two variables) give the local truncation error for the Improved Euler’s method as O ( h 3 ) For Improved Euler’s method , the global truncation error is O ( h 2 ) From a practical perspective, these results imply: With Euler’s method , the reduction of the stepsize by a factor of 0.1 gains one digit of accuracy With Improved Euler’s method , the reduction of the stepsize by a factor of 0.1 gains two digits of accuracy This is a significant improvement at only the cost of one additional function evaluation per step Joseph M. Mahaffy, h [email protected] i Lecture Notes – Numerical Methods for Differe — (34/39)
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Introduction Euler’s Method
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