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.

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.

References and Notes

1. Numerical Recipes in C  2