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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

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
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.