5.1.3 Steady-state solutions with operator adding or splitting

Suppose both operators $A$ and $B$ are zero,

$$A = B = 0$$ (214)

resulting in individually stationary solutions (eg. hydrostatic and radiative equilibrium),

$$q_A^{n+1} = q^{n} \enspace , \enspace \enspace q_B^{n+1} = q^{n} \enspace ,$$ (215)

which gives for both operator combination methods

$$q_{A+B}^{n+1} = q^{n} \enspace .$$ (216)

However, if the individual operators are non-zero but cancel each other,

$$B = -A \neq 0$$ (217)

the adding of the operator gives equilibrium Eq. (216), while operator splitting

$$q_A^{n,*} = q^{n} + \Delta t A(q^{n})$$

$$q_{A+-A}^{n+1} = q_A^{n,*} - \Delta t A(q_A^{n,*}) \enspace .$$ (218)

This is not necessarily a stationary solution. Here, operator adding is superior.