# Metodos numéricos en Mecánica de Fluidos

## Theory I

1-D Linear Convection equation is the simplest, most basic model that can be used to learn something about CFD.

$$\frac{\partial \phi}{\partial t} + u \frac{\partial \phi}{\partial x} = 0$$

From the definition of a derivative (and simply removing the limit), we know that:

$$\frac{\partial \phi}{\partial x}\approx \frac{ \phi(x+\Delta x)- \phi(x)}{\Delta x}$$

Our discrete equation, then, is:

$$\frac{ \phi_i^{n+1}- \phi_i^n}{\Delta t} + u \frac{ \phi_i^n - \phi_{i-1}^n}{\Delta x} = 0$$

Where $n$ and $n+1$ are two consecutive steps in time, while $i-1$ and $i$ are two neighboring points of the discretized $x$ coordinate.

If there are given initial conditions, then the only unknown in this discretization is $\phi_i^{n+1}$. We can solve for our unknown to get an equation that allows us to advance in time, as follows: $$\phi_i^{n+1} = \phi_i^n - u \frac{\Delta t}{\Delta x}( \phi_i^n- \phi_{i-1}^n)$$

Notice the only combination that appears is $$\mathrm{Co} := u \frac{\Delta t}{\Delta x}$$