  In this work, we numerically investigate the dynamics of a self-propelling
autophoretic Janus particle in a weakly viscoelastic fluid. The self-propulsion
is achieved by an asymmetry in the properties of the surface of the Janus
particle that drives a surface slip velocity and bulk flow. Here we investigate
the effect of viscoelasticity on this advection-diffusion problem over a range
of P\'eclet and Damk\"ohler numbers. Particles are found to swim faster, or
slower, in viscoelastic fluids, and we show how reaction and diffusion rates
affect the viscoelastic stresses that lead to changes in propulsion.
