finite_diff - Finite Difference Approximations Finite...

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Unformatted text preview: Finite Difference Approximations Finite difference methods are common tools for the numerical treatment of boundary value problems for both ordinary and partial differential equations. Here, we want to show how to define finite difference operators, more precisely central difference operators, that are used in these methods for the approximation of the first and second derivative of a function, u = u ( x ), defined on an interval [ a, b ]. Our tool is the Taylor expansion of a function about a given point. We first remark on the order symbol used in the approximations. Here we need only the notation O ( g ( x )) (read big-oh of g ( x )) which is commonly used. To say that a function f ( x ) is O ( g ( x )) as x 0 means simply that there is a constant M such that | f ( x ) | M | g ( x ) | for | x | sufficiently small. For example, we can write that e x- 1- x- x 2 / 2 is O ( x 3 ) as x 0. We usually write e x = 1 + x + x 2 / 2 + O ( x 3 ) to express this fact....
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finite_diff - Finite Difference Approximations Finite...

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