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66 4 Long Waves in a ChannelOn the basis of this, another approximation of the firs derivative of a function is:∂ f∂x≈f (x) − f (x − Δx)Δx(4.4)The third option is to take the sum of both Taylor series, yielding:∂ f∂x≈f (x + Δx) − f (x − Δx)2Δx(4.5)There are three different options of expressing the f rst derivative of a function interms of a finit difference.Fig. 4.1 Example of equidistant grid spacing. Distance is given by k · Δx,wherek is the cell indexand Δx is grid spacing4.1.2 Forward, Backward and Centred DifferencesWith the choice of equidistant grid spacing and index notation (Fig. 4.1), we canformulate the different finite-di ference forms of the firs derivative of a function as:∂ f∂x ≈ f k+1 − f kΔx(4.6)called forward difference, or∂ f∂x ≈ f k − f k−1Δx(4.7)called backward difference, or∂ f∂x ≈ f k+1 − f k−12Δx(4.8)called centred difference.4.1.3 Scheme for the Second DerivativeThe sum of the Taylor series (4.1) and (4.3) gives an approximation of the secondderivative of a function: