  In order to investigate the rheological properties of viscoelastic fluids by
mesoscopic hydrodynamics methods, we develop a multi-particle collision
dynamics (MPC) model for a fluid of harmonic dumbbells. The algorithm consists
of alternating streaming and collision steps. The advantage of the harmonic
interactions is that the integration of the equations of motion in the
streaming step can be performed analytically. Therefore, the algorithm is
computationally as efficient as the original MPC algorithm for Newtonian
fluids. The collision step is the same as in the original MPC method. All
particles are confined between two solid walls moving oppositely, so that both
steady and oscillatory shear flows can be investigated. Attractive wall
potentials are applied to obtain a nearly uniform density everywhere in the
simulation box. We find that both in steady and oscillatory shear flow, a
boundary layer develops near the wall, with a higher velocity gradient than in
the bulk. The thickness of this layer is proportional to the average dumbbell
size. We determine the zero-shear viscosities as a function of the spring
constant of the dumbbells and the mean free path. For very high shear rates, a
very weak ``shear thickening'' behavior is observed. Moreover, storage and loss
moduli are calculated in oscillatory shear, which show that the viscoelastic
properties at low and moderate frequencies are consistent with a Maxwell fluid
behavior. We compare our results with a kinetic theory of dumbbells in
solution, and generally find good agreement.
