Hybrid methods in planetesimal dynamics (I) : Description of a new
  composite algorithm
  The formation and evolution of protoplanetary systems, the breeding grounds
of planet formation, is a complex dynamical problem that involves many orders
of magnitudes. To serve this purpose, we present a new hybrid algorithm that
combines a Fokker-Planck approach with the advantages of a pure
direct-summation N-body scheme, with a very accurate integration of close
encounters for the orbital evolution of the larger bodies with a statistical
model, envisaged to simulate the very large number of smaller planetesimals in
the disc. Direct-summation techniques have been historically developped for the
study of dense stellar systems such as open and globular clusters and, within
some limits imposed by the number of stars, of galactic nuclei. The number of
modifications to adapt direct-summation N-body techniques to planetary dynamics
is not undemanding and requires modifications. These include the way close
encounters are treated, as well as the selection process for the "neighbour
radius" of the particles and the extended Hermite scheme, used for the very
first time in this work, as well as the implementation of a central potential,
drag forces and the adjustment of the regularisation treatment. For the
statistical description of the planetesimal disc we employ a Fokker-Planck
approach. We include dynamical friction, high- and low-speed encounters, the
role of distant encounters as well as gas and collisional damping and then
generalise the model to inhomogenous discs. We then describe the combination of
the two techniques to address the whole problem of planetesimal dynamics in a
realistic way via a transition mass to integrate the evolution of the particles
according to their masses.
