Differential Equation1.2

Differential Equation1.2 - Eulers Method Inaccurate and can...

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University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei Euler’s Method For t n = t 0 + n × h, 0 n (t last –t 0 )/h an approximate solution is given by u n+1 = u n + h*f(u n , t n ) Inaccurate and can be unstable: should not be used! Linear approximation is used to get the next point t n t n+1 t n+2 u Poor guess Real value u n h The main problem with Euler method is that only information from the beginning of the interval is used to extrapolate the value at the other side of the interval. In other words, only slope of the function is taken into account, the curvature is ignored .
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University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei Error in Euler’s method () ( ) ( ) ... dt t u d p! h ... dt t u d 2! h dt t du h u u p n p p 2 n 2 2 n n 1 n + + + + + = + Let’s use Taylor expansion to get the exact solution: Assuming that u n is exact, the estimation of local error due to the truncation is ) h ( O t , u hf u u 2 n n 1 n + + = + This is a local error. As it accumulates over the h -1
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Differential Equation1.2 - Eulers Method Inaccurate and can...

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