Particulate systems are widely used in various industries, such as cement, petrochemical, wastewater treatment, and pharmaceutical, where different types of particles are transported, mixed, stored, or segregated. In most of these processes, collisions of particles occur. In addition, it is important to know that most natural and man-made particles are non-spherical, ranging from blood composites to dust and aerosol particles in the air. To increase the efficiency of industrial processes or improve pharmaceutical applications such as targeted drug delivery, the physics of these non-spherical particle systems must be well known. However, understanding of the motion and interaction, i.e. collision, of arbitrarily shaped solids suspended in flows is still sparse as to date, most researchers have focused on spherical particles due to their simplified description of motion. Typically, shape factors are used to account for deviations from spherical shape, but these often fail to accurately predict particle motion. The superellipsoid shape formulation can be viewed as an extension of the spherical or ellipsoidal shape and can be used to represent spheres, ellipsoids, cylinder-like, and cubic particles by varying only five shape parameters. To accurately describe non-spherical particle systems, a fast, accurate, and stable non-spherical interaction model is required, which we target with our present work. In this study, we present a novel superellipsoid particle collision method that uses a fast, stable Newton-Raphson-based approach to predict collisions of non-spherical rigid superellipsoidal particles. The developed model is implemented in the open-source code OpenFOAM. We first validate the proposed numerical scheme using analytical results, followed by more complicated demonstrative collision examples. We show that the use of our novel superellipsoid collision model leads to sufficient accuracy of particle collision, which enables a further step towards modeling arbitrary shaped particulate systems.