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clase7handouts - Eulers method For the initial-value...

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Euler’s method For the initial-value problem y 0 = f ( t , y ) , y ( t 0 ) = y 0 , use the formulas t n + 1 = t n + h , y n + 1 = y n + hf ( t n , y n ) , to compute iteratively the points ( t 1 , y 1 ) , ..., ( t n , y n ) , using step size h . The piecewise-linear function connecting these points is the Euler approximation to the solution y ( t ) of the IVP for t 0 t t n . M.I. Bueno Differential Equations and Linear Algebra
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M.I. Bueno Differential Equations and Linear Algebra
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Example y 0 = t + y , y ( 0 ) = 1 [ 0 , 2 ] , h = 0 . 5 . 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 M.I. Bueno Differential Equations and Linear Algebra
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Example y 0 = t + y , y ( 0 ) = 1 [ 0 , 1 ] , h = 0 . 1 . n t n Approx. y n Real y ( t n ) Error | y ( t n ) - y n | Percent. erro 0 0 1 1 0 0 % 1 0.1 1.1 1.1103 0.0103 0 . 93 % 2 0.2 1.22 1.2428 0.0228 1 . 84 % 3 0.3 1.362 1.3997 0.0377 2 . 69 % 4 0.4 1.5282 1.5836 0.0554 3 . 5 % 5 0.5 1.721 1.7974 0.0764 4 . 25 % 6 0.6 1.9431 2.0442 0.1011 4 . 95 % 7 0.7 2.1974 2.3275 0.1301 5 . 59 % 8 0.8 2.4872 2.6511 0.1639 6 . 18 % 9 0.9 2.8159 3.0192 0.2033 6 . 73 % 10 1 3.1875 3.4366 0.2491 7 . 25 % M.I. Bueno Differential Equations and Linear Algebra
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.5 2 2.5 3 3.5 M.I. Bueno Differential Equations and Linear Algebra
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Error in Euler’s method How do we estimate the error produced by Euler’s method when we don’t have an analytic solution, the case of real interest for using numerical methods? There are two kinds of error that can arise: Roundoff error, Discretization error. M.I. Bueno Differential Equations and Linear Algebra
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Roundoff error Produced by rounding or chopping numbers at each stage in the computation. In practice, all calculators and computers have limitations in their computational accuracy. Roundoff errors can accumulate significantly after many steps. M.I. Bueno Differential Equations and Linear Algebra
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Discretization errors Results from the process itself. In Euler’s method, this process is the use of the linear approximation of the tangent line instead of the true solution curve in stepping from one value to the next. Local discretization error : the error that occurs in a single step as a result of the numerical formula. Global discretization error : accumulated error. M.I. Bueno Differential Equations and Linear Algebra
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For Euler’s method, this truncation error can be examined using Taylor’s Theorem.
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