Primary breakup of a liquid jet emanating from a simplex swirl atomizer is numerically studied in this work. The liquid-gas interface is tracked using a coupled Level-Set VOF method. In order to improve the curvature calculation, a dual grid approach is used. Level set function is solved on a grid that is finer than the grid used to solve the rest of the variables. This solver is developed using OpenFOAM libraries and is validated by estimating the pressure profile and spurious velocities across the interface of a static bubble. The validation test shows significant improvement in the performance of curvature estimation without significant increase in the computational cost. Additionally, it is found that curvature calculation improves with increase in grid refinement index. A parametric study to understand the effects of grid refinement index on the flow features of the simplex swirl atomizer primary jet is reported here.