Loading [MathJax]/extensions/TeX/boldsymbol.js
\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
Analysis of the Forward Euler scheme
\begin{equation*} [D_t^+ u = \dfc D_xD_x u]^n_q \end{equation*}
Inserting
u^n_q = A^n e^{ikq\Delta x}
leads to
A = 1 -4F\sin^2\left(
\frac{k\Delta x}{2}\right),\quad
F = \frac{\dfc\Delta t}{\Delta x^2}\mbox{ (mesh Fourier number)}
The complete numerical solution is
u^n_q = (1 -4F\sin^2 p)^ne^{ikq\Delta x},\quad
p = k\Delta x/2
Key spatial discretization quantity: the dimensionless p=\half k\Delta x
