Computational Physics Lectures: Partial differential equations

Truncation errors and stability

The following table summarizes the three methods.

*Scheme:*	*Truncation Error:*	*Stability requirements:*
Crank-Nicolson	$\mathcal{O}(\Delta x^2)$ and $\mathcal{O}(\Delta t^2)$	Stable for all $\Delta t$ and $\Delta x$ .
Backward Euler	$\mathcal{O}(\Delta x^2)$ and $\mathcal{O}(\Delta t)$	Stable for all $\Delta t$ and $\Delta x$ .
Forward Euler	$\mathcal{O}(\Delta x^2)$ and $\mathcal{O}(\Delta t)$	$\Delta t\leq \frac{1}{2}\Delta x^2$