Schematic diagram for the location of our computational domain and transition from a hydrostatic solution to Roche lob overflow. The top panel depicts the shape of the Roche-lobe equipotential surfaces, along with the approximate locations of our computational domains (rectangles) around L_in (red cross) and L_out (blue cross). The arrows indicate the direction of overflowing gas. The diagram is drawn not to scale. The bottom panels depict the density change, with denser regions shown in darker colors, during the transition from a contact binary in hydrostatic equilibrium to overflow by imposing an outflow boundary condition (BC) along the right x boundary. The dashed diagonal orange lines depict the shape of the RL.