  We have performed Smoothed Particle Hydrodynamics (SPH) simulations to study
the time evolution of one and two protoplanets embedded in a protoplanetary
accretion disc. We investigate accretion and migration rates of a single
protoplanet depending on several parameters of the protoplanetary disc, mainly
viscosity and scale height. Additionally, we consider the influence of a second
protoplanet in a long time simulation and examine the migration of the two
planets in the disc, especially the growth of eccentricity and chaotic
behaviour. One aim of this work is to establish the feasibility of SPH for such
calculations considering that usually only grid-based methods are adopted. To
resolve shocks and to prevent particle penetration, we introduce a new approach
for an artificial viscosity, which consists of an additional artificial bulk
viscosity term in the SPH-representation of the Navier-Stokes equation. This
allows for an accurate treatment of the physical kinematic viscosity to
describe the shear, without the use of an artificial shear viscosity.
