$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Stability criterion in 3D
$$
\Delta t \leq \frac{1}{c}\left( \frac{1}{\Delta x^2} +
\frac{1}{\Delta y^2} + \frac{1}{\Delta z^2}\right)^{-\halfi}
$$
For \( c^2=c^2(\xpoint) \) we must use
the worst-case value \( \bar c = \sqrt{\max_{\xpoint\in\Omega} c^2(\xpoint)} \)
and a safety factor \( \beta\leq 1 \):
$$
\Delta t \leq \beta \frac{1}{\bar c}
\left( \frac{1}{\Delta x^2} +
\frac{1}{\Delta y^2} + \frac{1}{\Delta z^2}\right)^{-\halfi}
$$
