Im trying to do some numerical analysis using finite differences method, but have some questions:
There is an analitical solution $u(x,t)$ and the numerical estimation $U_n(x,t)$. I want to compare their spatial variation and in order to do it I take the Taylor series like this:
$$U_n(x_j+\Delta x,t)=U_n(x_j,t)+\Delta x\dfrac{\partial u}{\partial x}\bigg|_{(x_j,t)}+\frac{(\Delta x)^2}{2!}\dfrac{\partial^2 u}{\partial x^2}\bigg|_{(x_j,t)}$$
Now i can make
$$\frac{U_n(x_j+\Delta x,t)-U_n(x_j,t)}{\Delta x}-\dfrac{\partial u}{\partial x}\bigg|_{(x_j,t)}=\frac{\Delta x}{2!}\dfrac{\partial^2 u}{\partial x^2}\bigg|_{(x_j,t)}$$ And later i estimated this difference by $C(\Delta x)$, where C is a bound for $\bigg|\dfrac{\partial^2 u}{\partial x^2}\bigg|$
The problem is that i check in some books and they say that the estimation is $C(\Delta x+\Delta t)$ but i cant noticed why....