5.1.4 Linear stability of operator adding or splitting

If both operators $\bgroup\color{DEFcolor}$A$\egroup$ and $\bgroup\color{DEFcolor}$B$\egroup$ are linear with amplification factors indicating stability

$\begin{displaymath} V_A \le 1 \enspace , \enspace \enspace V_B \le 1 \end{displaymath}$

(219)

then the amplification factor of the scheme combined by operator splitting is

$\begin{displaymath} V_{A+B} = V_A V_B \le 1 \end{displaymath}$

(220)

implying stability of the combined scheme.

The adding of the operators requires a re-analysis of the combined scheme. It might be stable or not, even if Eq. (219) applies.

The individual analysis of a non-linear hydrodynamics and a complex non-local radiation transport scheme is difficult enough. It is convenient to have a combination method, that - at least in some simple cases - guarantees the stability of the combination

Here, operator splitting is superior.