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lectfiniteDifference - Chapter 11 Finite Difference...

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Chapter 11 Finite Difference Approximation of Derivatives 11.1 Introduction The standard definition of derivative in elementary calculus is the following u 0 ( x ) = lim Δ x 0 u ( x + Δ x ) - u ( x ) Δ x (11.1) Computers however cannot deal with the limit of Δ x 0, and hence a discrete analogue of the continuous case need to be adopted. In a discretization step, the set of points on which the function is defined is finite, and the function value is available on a discrete set of points. Approximations to the derivative will have to come from this discrete table of the function. Figure 11.1 shows the discrete set of points x i where the function is known. We will use the notation u i = u ( x i ) to denote the value of the function at the i -th node of the computational grid. The nodes divide the axis into a set of intervals of width Δ x i = x i +1 - x i . When the grid spacing is fixed, i.e. all intervals are of equal size, we will refer to the grid spacing as Δ x . There are definite advantages to a constant grid spacing as we will see later. 11.2 Finite Difference Approximation The definition of the derivative in the continuum can be used to approximate the derivative in the discrete case: u 0 ( x i ) u ( x i + Δ x ) - u ( x i ) Δ x = u i +1 - u i Δ x (11.2) where now Δ x is finite and small but not necessarily infinitesimally small, i.e. . This is known as a forward Euler approximation since it uses forward differencing. 77
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78 CHAPTER 11. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES x i-1 x i x i+1 Figure 11.1: Computational grid and example of backward, forward, and central approximation to the derivative at point x i . The dash-dot line shows the centered parabolic interpolation, while the dashed line show the backward (blue), forward (red) and centered (magenta) linear interpolation to the function. Intuitively, the approximation will improve, i.e. the error will be smaller, as Δ x is made smaller. The above is not the only approximation possible, two equally valid approximations are: backward Euler: u 0 ( x i ) u ( x i ) - u ( x i - Δ x ) Δ x = u i - u i - 1 Δ x (11.3) Centered Difference u 0 ( x i ) u ( x i + Δ x ) - u ( x i - Δ x ) x = u i +1 - u i - 1 x (11.4) All these definitions are equivalent in the continuum but lead to different approx- imations in the discrete case. The question becomes which one is better, and is
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11.3. TAYLOR SERIES 79 there a way to quantify the error committed. The answer lies in the application of Taylor series analysis. We briefly describe Taylor series in the next section, before applying them to investigate the approximation errors of finite difference formulae. 11.3 Taylor series Starting with the identity: u ( x ) = u ( x i ) + Z x x i u 0 ( s ) d s (11.5) Since u ( x ) is arbitrary, the formula should hold with u ( x ) replaced by u 0 ( x ), i.e., u 0 ( x ) = u 0 ( x i ) + Z x x i u 00 ( s ) d s (11.6) Replacing this expression in the original formula and carrying out the integration (since u ( x i ) is constant) we get: u ( x ) = u ( x i ) + ( x - x i ) u 0 ( x i ) + Z x x i Z x x i u 00 ( s ) d s d s (11.7) The process can be repeated with u 00 ( x ) = u 00 ( x i ) + Z x x i u 000 ( s ) d s (11.8) to get: u ( x ) = u ( x i ) + ( x - x i ) u 0 ( x i ) + ( x - x i ) 2 2!
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