Results at 23.3 Gyr for the two-ring test. Two rings rotate in opposite directions around a 1 × 10¹¹ M_⊙ mass located at (0,0). When the frame of reference rotates (all but the standard GEAR code), it rotates around the point (100,0). The points should continue to orbit at 50 kpc and 100 kpc represented by two black rings. The standard GEAR code in a non-rotating frame is shown with blue crosses. In a rotating frame of reference, the results of calculating the forces at every time step is shown with yellow squares, using the half time step velocity (Mid-v) as green points, and the iteration scheme (It-v) with red triangles. Only every 20th point is shown for clarity.