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A fully-simultaneous method is based on the assumption that there is no precise hierarchy between viscous and inviscid solutions. This statement has been proved with the triple-deck boundary layer theory (Stewartson, 1974). The simultaneous solution of the two sets of equations is performed by a Newton-like iteration process after assuming linearizations of the type Eq. 8 and 9 (direct method).
The response to small perturbations of a (generally) non-linear system is
linear. Perturbations
The simultaneous solution of Eqs. 16 and 17 at the current point
The solution of Eq. 18 and 19 involves the inversion of the operator
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