Study guide: Finite difference schemes for diffusion processes

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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

$u'(t)=-\dfc u(t)$ .