The coarse-mesh finite difference (CMFD) method is one of the most widely used methods for accelerating the convergence of numerical transport solutions. However, in some situations, iterative methods using CMFD can become unstable and fail to converge. We present and evaluate three different modifications of the CMFD scheme that provide enhanced stability: multiple transport sweeps, artificial diffusion, and relaxing the flux update. We present the Fourier analysis on each of these schemes for an idealized problem to characterize the stability and rate of convergence for both fixed-source and fission-source problems. Comparisons of the effectiveness of these methods are also performed numerically for a variety of benchmark boiling water reactor and pressurized water reactor problems using the Consortium for Advanced Simulation of Light Water Reactors neutronics code MPACT. We demonstrate a means of stabilizing CMFD by modifying the diffusion coefficient to make the iteration behave more like the partial-current CMFD (pCMFD) method, which is unconditionally stable, and show through a sequence of numerical experiments that the CMFD method performs similarly to the pCMFD method for the selected benchmark problems. We also show, both theoretically and experimentally, that modifying the diffusion coefficient in the CMFD equations is similar to underrelaxing the scalar flux update. The theoretical and experimental results show that many of the known techniques for stabilizing CMFD are fundamentally very closely related.