Question

Find the first-order forward difference approximation of \((\frac{\partial u}{\partial x})_{i,j}\) using the Taylor series expansion.

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

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

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

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

The question was posed to me in an online quiz.

I want to ask this question from Finite Difference Method in portion Finite Difference Methods of Computational Fluid Dynamics

Answer 1

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

The best I can explain: To get the first-order forward difference approximation,

The Taylor series expansion of ui+1,j is

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

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