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Self-interacting dark matter (SIDM) models have the potential to solve the small-scale problems that arise in the cold dark matter paradigm. Simulations are a powerful tool for studying SIDM in the context of astrophysics, but it is numerically challenging to study differential cross-sections that favour small-angle scattering, as in light-mediator models. Here, we present a novel approach to model frequent scattering based on an effective drag force, which we have implemented into the N-body code gadget-3. In a range of test problems, we demonstrate that our implementation accurately models frequent scattering. Our implementation can be used to study differences between SIDM models that predict rare and frequent scattering. We simulate core formation in isolated dark matter haloes, as well as major mergers of galaxy clusters and find that SIDM models with rare and frequent interactions make different predictions. In particular, frequent interactions are able to produce larger offsets between the distribution of galaxies and dark matter in equal-mass mergers.