# Partial Differential Equation¶

## Forward Time Centered Space¶

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

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

FTCS is an explicit method and is not stable.

## Lax Method¶

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

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.

## 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¶

 [NumericalRecipes] (1, 2) Numerical Recipes in C

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