Hydrodynamical simulations of semi-detached , polytropic binary stars are presented in an effort to study the onset and stability of dynamical mass transfer events . Initial , synchronously rotating equilibrium models are constructed using a self-consistent-field technique and then evolved with an Eulerian hydrodynamics code in a fully self-consistent manner . We describe code improvements introduced over the past few years that permit us to follow dynamical mass-transfer events through more than 30 orbits . Mass-transfer evolutions are presented for two different initial configurations : A dynamically unstable binary with initial mass ratio ( donor/accretor ) q _ { 0 } = 1.3 that leads to a complete merger in \sim 10 orbits ; and a double-degenerate binary with initial mass ratio q _ { 0 } = 0.5 that , after some initial unstable growth of mass transfer , tends to separate as the mass-transfer rate levels off .