I would like to know on which method (e.g. Douglas-Peucker line simplification) the modules qgis:simplifygeometries and qgis:smoothgeometry are built on.
I cannot find further information on these modules.
Simplifying and smoothing operations are related to the QgsGeometry() Class: this means that you can run them when dealing with the geometry of the current feature.
As far as I know, the Simplify geometries algorithm literally simplifies the current geometry by reducing the number of vertices on the basis of a tolerance value (so, there isn't any particular method applied).
Instead, the Smooth geometry algorithm is based on the Chaikin algorithm, as reported in the smooth() module:
Smooths a geometry by rounding off corners using the Chaikin algorithm.
This operation roughly doubles the number of vertices in a geometry.
it seems that there is an explicit reference to the Douglas-Peucker algorithm. More in detail, from the first link above:
Simplifies a linestring (sequence of points) using the standard Douglas-Peucker algorithm.