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GATE 2020 CE (CE1) – Question 37

Engineering Mathematics · Numerical Methods: Error analysis, algebraic equations, interpolation, differentiation, integration, ODEs · 2 marks · Multiple choice

For a continuous function f, choose the fourth-order central finite-difference approximation to $f^{\prime\prime\prime}(x_i)$, with step h.

  1. $[-f_{i+3}+8f_{i+2}-13f_{i+1}+13f_{i-1}-8f_{i-2}+f_{i-3}]/(8h^3)$
  2. $[f_{i+3}-8f_{i+2}-13f_{i+1}+13f_{i-1}+8f_{i-2}+f_{i-3}]/(8h^3)$
  3. $[-f_{i+3}-8f_{i+2}-13f_{i+1}+13f_{i-1}+8f_{i-2}-f_{i-3}]/(8h^3)$
  4. $[f_{i+3}-8f_{i+2}-13f_{i+1}+13f_{i-1}+8f_{i-2}-f_{i-3}]/(8h^3)$

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Correct answer: (A) $[-f_{i+3}+8f_{i+2}-13f_{i+1}+13f_{i-1}-8f_{i-2}+f_{i-3}]/(8h^3)$

Explanation

The antisymmetric stencil has weights 1, −8, 13, −13, 8, −1 at offsets −3, −2, −1, 1, 2, 3, divided by $8h^3$. Taylor expansion cancels all terms through sixth order except the third derivative. The next term gives an $O(h^4)$ error. **A**. Applying it to $f(x)=x^3$ gives 6, confirming its sign.