Surface discontinuities, such as cylindrical protrusions or dents, are commonly found on artificial aircraft or vehicle surfaces. Despite being electrically small in size, their sheer quantity can significantly affect the scattering characteristics of the target. Due to the large contrast between the surface discontinuity and surrounding geometry, the efficient scattering analysis of these targets is technically challenging in both mesh generation and numerical simulation. In this communication, two novel basis functions are presented based on the discontinuous Galerkin integral equation (DGIE) framework. Using the so-called extended half-RWG and modified top junction basis functions, the cylindrical surface discontinuities with different or variable height-radius ratios can be efficiently discretized without requiring mesh conformality. Extensive numerical simulations of surface discontinuities on a low-RCS testbed are performed to verify the accuracy, flexibility, and robustness of the proposed method.