calculation took $\approx 5$ months on a 170 MHz Sun Ultra workstation with a GRAvity-PipE (GRAPE) board used to calculate the gravitational forces and neighbouring SPH particles. The evolution with the PBE code took a few seconds!

Although the SPH calculation did not run to completion, we can compare the evolution as the binary’s mass increases by a factor of 6 (Figure \[bate.testcase1\]). Generally, there is good agreement between the PBE and SPH codes. The mass ratio is predicted to within 5% over the entire evolution and the separation to within 15%. In fact, as discussed in [@Bate2000], the small differences between the PBE and SPH results reflect unphysical treatment of the circumstellar discs by the SPH code rather than a problem with the PBE code. For example, the slower rate of increase of the mass ratio initially is due to the circumsecondary disc not being resolved correctly in the SPH calculation, and the larger separation when $M_{\rm acc}\simgreat 3$ is due to unphysically-rapid viscous evolution of the circumstellar discs which transfers angular momentum into the binary’s orbit too quickly. The greatest difference between the PBE and SPH results is that the PBE code predicts that a circumbinary disc should 