For my computational fluid dynamics course I've been given a multi-step problem to solve. I've tried solving this numerous times but I seem to be confused about the set up. For now, I'll just post the first part of the question as this seems to be the part I'm having the most difficulty.

Problem:

For a given function u=u(x,t), start with Taylor series to implement the forward finite difference expression for the heat conduction equation in non-dimensional form

$\displaystyle \frac{\partial u}{\partial t}=\frac{\partial ^2 u}{\partial x^2}$

by using the following steps.

Obtain the first three terms truncated for each derivative in the above equation. The answer should be in terms of step sizes $\displaystyle \Delta t$ and $\displaystyle \Delta x$ and the derivatives of u with respect to t and x. Use subscript i for x and superscript n for t to represent the nodes.

Attempt/Confusion:

The first 3 terms of the Taylor series

$\displaystyle u(x_0+\Delta x)=u(x_0)+\frac{\Delta x}{1!}u'(x_0,u(x_0))+\frac{\Delta x^2}{2!}u''(x_0,u(x_0))+O(\Delta x)^2$

I'm not quit sure what they mean by "obtain the first three terms truncated for each derivative". I know that truncated means I don't worry about the higher-level terms.

The forward finite difference

$\displaystyle \frac{\partial u}{\partial t}=\frac{\partial ^2 u}{\partial x^2}\rightarrow\frac{u_i^{n+1}-u_i^n}{\Delta t}=\frac{u_{i+1}^n-2u_i^n+u_{i-1}^n}{(\Delta x)^2}$

My first thought was to substitute in the forward finite difference expression for $\displaystyle \frac{\partial ^2 u}{\partial x^2}$ into the Taylor series expression but then I realized that the $\displaystyle \Delta$'s would cancel. Is my thinking correct? What am I screwing up here?