Question

Find the second-order accurate finite difference approximation of the first derivative of the velocity component (u) in the x-direction using the Taylor series expansion. (Note: i and j are in the x and y-direction respectively).

(a) \(\frac{u_{i,j+1}-u_{i,j-1}}{\Delta x}\)

(b) \(\frac{u_{i+1,j}-u_{i-1,j}}{\Delta x}\)

(c) \(\frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x}\)

(d) \(\frac{u_{i,j+1}-u_{i,j-1}}{2\Delta x}\)

I have been asked this question in examination.

This interesting question is from Finite Difference Method in chapter Finite Difference Methods of Computational Fluid Dynamics

Answer 1

The correct option is (c) \(\frac{u_{i+1,j}-u_{i-1,j}}{2\Delta x}\)

For explanation: The only second-order accurate finite difference approximation of the first derivative is the central difference. For getting the central difference term,

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

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

To get \((\frac{\partial u}{\partial x})_{i,j^,}\)

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

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