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.
$\includegraphics[width=5.2cm]{images/pde1d_solve_heat_cosshortc_simple_i10n1dt02.ps}$ $\includegraphics[width=5.2cm]{images/pde1d_solve_heat_cosshortc_simple_i10n1dt04.ps}$ $\includegraphics[width=5.2cm]{images/pde1d_solve_heat_cosshortc_simple_i10n1dt06.ps}$

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.