Partial Differential Equations

Forward Time Centered Space

For $\frac{df}{dt}=-v\frac{df}{dx}$, we write down the finite difference form ¹

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

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

$$\frac{|v|\mathrm{\Delta }t}{\mathrm{\Delta }x}\le 1,$$

which is the Courant-Fridriches-Lewy stability criterion.

Staggered Leapfrog

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

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

Two-Step Lax-Wendroff Scheme

Fully Implicit

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

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.

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

References and Notes