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.

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

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

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