Question

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 1

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 –\((\frac{\partial^2 u}{\partial x^2})_{i,j}\frac{\Delta x}{2}\).