Loading [MathJax]/extensions/TeX/boldsymbol.js
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
Lowering the order of the derivative in the correction term
- C^n contains u''
- Can discretize u'' (requires u^{n+1} , u^n , and u^{n-1} )
- Can also express u'' in terms of u' or u
u' = -au,\quad\Rightarrow\quad u''=-au'=a^2u\tp
Result for u''=a^2u : apply Forward Euler to a perturbed ODE,
\begin{equation}
u' = -\hat au ,\quad \hat a = a(1 - {\half}a\Delta t),
\tag{31}
\end{equation}
to make a second-order scheme!
