Abstract In this study, the GDQ method was used to simulate a specific flow regime, Taylor vortex flow, of the motion of fluids between two concentric cylinders with rotating inner cylinder and stationary outer cylinder. An approach combining the SIMPLE strategy and GDQ discretization based on non-staggered mesh was proposed to solve the time-dependent, three-dimensional incompressible Navier-Stokes equations in primitive variable form. The numerical solution obtained has the accuracy of second-order in temporal discretization and high-order in spatial discretization. Also, this numerical code may allow the direct numerical simulations for the various regimes of Couette-Taylor flow problem. The performance of this approach was studied through a test case of Taylor vortex flow. The reported numerical results were compared with those from others. For this approach, accurate numerical results can be obtained by using fewer grid points compared with low-order methods. As a consequence, the computational effort can be greatly reduced.