The flow-induced orientation of liquid crystalline polymers (LCPs) would substantially affect the mechanical and optical properties of the finished product. In order to investigate the director orientation of nematic polymers in complex geometrical configurations, an efficient and accurate algorithm for the fluid particle tracking technique as well as an iterative scheme are introduced in this article. The simplified Leslie-Ericksen equations were employed and calculated on the basis of the high viscosity approximation. Director orientations predicted by other numerical schemes were found to be polluted by the artificial cross-wind diffusion of improper numerical upwinding methods, especially in the recirculation regions. This was circumvented in the proposed algorithm by integrating the director transport equation along streamlines with a fourth-order Runge-Kutta method. Therefore, the director orientation in a steady recirculation zone could be visualized and the numerical computation was also confirmed to be more stable. To demonstrate the power of the numerical method as well as the effects of the director orientation parameter and anisotropic viscosities on the director orientation, two examples, i.e., expansion flow and contraction flow, were studied. The rheological parameter lambda was found to have a strong influence on the director orientation, while beta1 and beta2 were found to affect the pressure drop across the flow field.