--- title: "Derive forward, backward, central differences" subject: "Applied Computational Fluid Dynamics" chapter: 3 key: "semester_7/applied_computational_fluid_dynamics:taylor-series-finite-differences" source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/taylor-series-finite-differences site: BCT Engineering --- # Derive forward, backward, central differences Applied Computational Fluid Dynamics · Chapter 3: Discretization Methods - Asked in: year not given, 6 marks - Video: [Forward, Backward, and Central Difference Method](https://www.youtube.com/watch?v=Jqa-aFE9-GI) — Alex Malt - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/taylor-series-finite-differences - Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-3 ## Question Using Taylor series expansion, derive the forward, backward and central difference approximations for the first derivative $df/dx$ and state the truncation error and order of accuracy of each. ## Short answer From Taylor series: forward difference (f_{i+1}−f_i)/h, truncation error O(h); backward (f_i−f_{i-1})/h, O(h); central (f_{i+1}−f_{i-1})/(2h), O(h²). Halving h halves error for first order, quarters for second order. ## Answer The **finite difference method (FDM)** replaces derivatives by algebraic differences of function values at grid points, spaced $h=\Delta x$ apart. The basis is the Taylor series. #### Taylor expansions about $x_i$ $$ f_{i+1}=f_i+h f'_i+\frac{h^2}{2!}f''_i+\frac{h^3}{3!}f'''_i+\cdots $$ $$ f_{i-1}=f_i-h f'_i+\frac{h^2}{2!}f''_i-\frac{h^3}{3!}f'''_i+\cdots $$ #### Forward difference Rearrange the first series for $f'_i$: $$ f'_i=\frac{f_{i+1}-f_i}{h}-\frac h2f''_i-\frac{h^2}{6}f'''_i-\cdots $$ $$ \boxed{f'_i\approx\frac{f_{i+1}-f_i}{h}},\qquad \text{error}=-\frac h2f''_i=O(h) $$ #### Backward difference Rearrange the second series: $$ f'_i=\frac{f_i-f_{i-1}}{h}+\frac h2f''_i-\cdots $$ $$ \boxed{f'_i\approx\frac{f_i-f_{i-1}}{h}},\qquad \text{error}=+\frac h2f''_i=O(h) $$ #### Central difference Subtract the second series from the first; the even-order terms cancel: $$ f_{i+1}-f_{i-1}=2hf'_i+\frac{h^3}{3}f'''_i+\cdots $$ $$ \boxed{f'_i\approx\frac{f_{i+1}-f_{i-1}}{2h}},\qquad \text{error}=-\frac{h^2}{6}f'''_i=O(h^2) $$ #### Summary | Scheme | Formula | Truncation error | Order | |---|---|---|---| | Forward | $(f_{i+1}-f_i)/h$ | $-\tfrac h2f''$ | 1st | | Backward | $(f_i-f_{i-1})/h$ | $+\tfrac h2f''$ | 1st | | Central | $(f_{i+1}-f_{i-1})/2h$ | $-\tfrac{h^2}{6}f'''$ | 2nd | Halving $h$ halves the error of the first-order schemes but reduces the central-difference error to one quarter. Forward and backward differences are used at boundaries and in upwind schemes, where only one side is available. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The forward difference (f_{i+1} - f_i)/h has truncation error and order: A. -h^2/6 f''', second order B. -(h/2) f'', first order C. +h/2 f'', second order D. Zero error Answer: B. Rearranging the Taylor series leaves -h/2 f'', which is O(h). (Key point: Forward difference error) 2. How is the central difference obtained? A. Adding the two Taylor series B. Using only the first series C. Subtracting the backward series from the forward one so even-order terms cancel D. Multiplying the series Answer: C. Subtraction gives f_{i+1} - f_{i-1} = 2h f' + h^3/3 f''' + ..., so the h^2 term cancels. (Key point: Central difference derivation) 3. Halving h reduces the central-difference error by a factor of: A. 2 B. 8 C. 1 D. 4 Answer: D. It is O(h^2), so halving h reduces the error to one quarter. (Key point: Central difference accuracy) 4. Compared with the forward error, the backward difference error is: A. Same order but opposite sign, +h/2 f'' B. Second order C. Zero D. Same sign and larger Answer: A. Backward error is +(h/2) f'', first order and opposite in sign. 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