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2.2.6 Parabolic PDE: stability

Figure 2: Stability of Euler scheme for heat equation: $\Delta tK/\Delta x^2$=0.2, 0.4, 0.6.
\includegraphics[width=5.2cm]{images/} \includegraphics[width=5.2cm]{images/} \includegraphics[width=5.2cm]{images/}
Examples for a simple \bgroup\color{DEFcolor}$(1,0,1,0,\ldots)$\egroup initial condition (red) are displayed in Fig. 2. The
$\displaystyle \mbox{{\color{HIGH1color}criterion for stability}:\index{stability!of parabolic PDE}}$ $\textstyle \hspace{2cm}$ $\displaystyle \Delta t/\Delta x^2 < 1/(2K)$ (86)
$\displaystyle \mbox{{\color{HIGH1color}criterion for positivity}:}$ $\textstyle \hspace{2cm}$ $\displaystyle \Delta t/\Delta x^2 < 1/(4K)$ (87)
are comparable to the case of a ODE.