# Partial Differential Equations

Difference equations are fun!

## Forward Time Centered Space

For $\frac{d f}{d t} = - v \frac{ d f }{ dx }$, we write down the finite difference form ^{1}

FTCS is an explicit method and is not stable.

## Lax Method

Change the term $f(t_n, x_i)$ in FTCS to $( f(t_n, x_{i+1}) + f(t_n, x_{i-1}) )/2$ ^{1}.

Stability condition is

which is the Courant-Fridriches-Lewy stability criterion.

## Staggered Leapfrog

It’s kind of a Centered Space Centered Time method.

## Two-Step Lax-Wendroff Scheme

## Fully Implicit

It is called implicity because we can not simply iterate over the formula to get the solutions as like for the explicit method.

## Crank-Nicholson

Crank-Nicholson is a average of the explicit and fully implicit method.