In hydrodynamic lubrication problems, the presence of step structures on the surface can cause discontinuities in the film thickness. This paper proposes two models for solving the two-dimensional Reynolds equation with film thickness discontinuity using the finite difference method (FDM). In Model I, the film thickness variable is defined at the center of the mesh grids, allowing the Reynolds equation to be reformulated in a weak form that eliminates the singularity of film thickness discontinuity and satisfies the flow continuity condition at the film thickness discontinuity region. By considering the step boundary on the surface as the interface, Model II is constructed based on the immersed boundary method, turning the hydrodynamic lubrication problem into a classical interface problem. The jump conditions across the interface are derived in accordance with the continuous flow requirement. A phase-field function is adopted to describe the interface on the uniform rectangular mesh grids. Numerical experiments are conducted to assess the accuracy and capabilities of the two proposed models for analyzing a step-dimple-textured sealing. The results demonstrate that both modified FDM models can effectively address the thickness discontinuity issue. Model II achieves second-order accuracy for the pressure distribution when dealing with curved interfaces based on Cartesian grids, whereas Model I demonstrates first-order accuracy. Moreover, the performance of Model II is further assessed by simulating lubrication problems with complex groove shapes and the results indicate its robustness and flexibility in addressing thickness discontinuity problems with complex curve interface.