The algorithm should follow Andreasen et al. (2018) as already implemented in simult_.m for orders 2 and 3. Should be done in the C++ routines of dynare_simul_ as discussed in 1e92e308 (comment 8364)
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Actually, the manual says that order 2 is implemented using Kim, Kim, Schaumburg and Sims (2008) and order 3 uses Andreasen, Fernández-Villaverde and Rubio-Ramírez (2013).
KKSS were first. Andreasen et al (2018) (the 2013 working paper has been published in ReStud) nests KKSS at order 2 and outlines how to generalize to higher order.
Related: we need to investigate the speed differences, see #1773 (closed)
It may be desirable to use straight Fortran code instead of Dynare++ routines.