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Using the Taylor series expansion, What is the first term of the truncation error of the finite difference equation \((\frac{\partial u}{\partial x})_{i,j}=\frac{u_{i+1,j}-u_{i,j}}{\Delta y}\)?(a) \(-(\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}\)(b) \((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{3}\)(c) \(-(\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{3}\)(d) \((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}\)This question was posed to me during a job interview.My doubt stems from Finite Difference Method topic in section Finite Difference Methods of Computational Fluid Dynamics

Answer»

The CORRECT choice is (a) \(-(\frac{\partial^2 U}{\partial X^2})_{i,j}\frac{\DELTA x}{2}\)

EXPLANATION: The Taylor series expansion of ui+1,j is

\(u_{i+1,j}=u_{i,j}+(\frac{\partial u}{\partial x})_{i,j} \Delta x+(\frac{\partial ^2 u}{\partial x^2})_{i,j}\frac{(\Delta x)^2}{2}+⋯\)

\(\frac{u_{i+1,j}-u_{i,j}}{\Delta x}=(\frac{\partial u}{\partial x})_{i,j}+(\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}+⋯\)

\((\frac{\partial u}{\partial x})_{i,j}=\frac{u_{i+1,j}-u_{i,j}}{\Delta x}-(\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}-…\)

The term –\((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}-… \)is truncated. So, the first term of truncation error is –\((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}\).


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