$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Numerical dispersion relation
- How close is \( \tilde\omega \) to \( \omega \)?
- Can solve for an explicit formula for \( \tilde\omega \)
$$
\tilde\omega = \frac{2}{\Delta t}
\sin^{-1}\left( C\sin\left(\frac{k\Delta x}{2}\right)\right)
$$
- \( \omega = kc \) is the analytical dispersion relation
- \( \tilde\omega = \tilde\omega(k, c, \Delta x, \Delta t) \) is the
numerical dispersion relation
- Speed of waves: \( c=\omega/k \), \( \tilde c = \tilde\omega/k \)
- The numerical wave component has a wrong, mesh-dependent speed
