Partial Differential Equations

Forward Time Centered Space

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

$$ \frac{f(t_{n+1}, x_i ) - f(t_n, x_i)}{ \Delta t } = - v \frac{ f(t_n, x_{i+1}) - f(t_n, x_{i-1}) }{ 2\Delta x }. $$

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

$$ \frac{ \lvert v \rvert \Delta t }{ \Delta x } \leq 1, $$

which is the Courant-Fridriches-Lewy stability criterion.

Staggered Leapfrog

$$ \frac{f(t_{n+1}, x_i) - f(t_{n-1}, x_i)}{2 \Delta t} = -v \frac{ f(t_n, x_{i+1} ) - f(t_n, x_{i-1} ) }{ 2\Delta x} $$

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

Two-Step Lax-Wendroff Scheme

Fully Implicit

$$ \frac{ f( t_{n+1} , x_i ) - f( t_{n} , x_i ) }{ \Delta t } = - v \frac{ f(t_{n+1}, x_{i+1}) - f(t_{n+1}, x_{i-1}) }{ 2\Delta x }. $$

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.

$$ \frac{ f( t_{n+1} , x_i ) - f( t_{n} , x_i ) }{ \Delta t } = - \frac{v}{2} \frac{ \left(f(t_{n+1}, x_{i+1}) - f(t_{n+1}, x_{i-1}) \right) + \left( f(t_{n}, x_{i+1}) - f(t_{n}, x_{i-1}) \right)}{ 2\Delta x }. $$

References and Notes