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5.1.1 Operator adding versus splitting

Let the time evolution of \bgroup\color{DEFcolor}$q$\egroup be determined by two operators \bgroup\color{DEFcolor}$A$\egroup and \bgroup\color{DEFcolor}$B$\egroup according to

\frac{\mathrm{d} q}{\mathrm{d} t} = A(q) + B(q)
\enspace .
\end{displaymath} (209)
Suppose there are separate numerical schemes available that allow to compute the individual updates
q_A^{n+1} = q^{n} + \Delta t  A(q^{n})
\enspace , \enspace \enspace
q_B^{n+1} = q^{n} + \Delta t  B(q^{n})
\enspace .
\end{displaymath} (210)
Now, the two schemes could be combined in two ways, e.g. by operator adding
q_{A+B}^{n+1} = q^{n} + \Delta t  A(q^{n}) + \Delta t  B(q^{n})
\end{displaymath} (211)
or Godunov operator splitting
$\displaystyle q_A^{n,*}$ $\textstyle =$ $\displaystyle q^{n} + \Delta t  A(q^{n})$  
$\displaystyle q_{A+B}^{n+1}$ $\textstyle =$ $\displaystyle q_A^{n,*} + \Delta t  B(q_A^{n,*}) \enspace .$ (212)
The results are generally not the same. Both methods have advantages/drawbacks.