$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
Observations
- The key spatial discretization parameter is the dimensionless \( p=\half k\Delta x \)
- The key temporal discretization parameter is the dimensionless \( F = \dfc\Delta t/\Delta x^2 \)
- Important: \( \Delta t \) and \( \Delta x \) in combination with \( \dfc \) and \( k \)
determine accuracy
- Crank-Nicolson gives oscillations and not much damping of short waves
for increasing \( F \)
- These waves will manifest themselves as high frequency
oscillatory noise in the solution
- Steep solutions will have short waves with significant (visible) amplitudes
- All schemes fail to dampen short waves enough
The problems of correct damping for \( u_t = u_{xx} \) is partially
manifested in the similar time discretization schemes for \( u'(t)=-\dfc u(t) \).
