Numerical schemes for a fully nonlinear coagulation-fragmentation model coming from wave kinetic theory

This article introduces a novel numerical approach, based on Finite Volume Techniques, for studying fully nonlinear coagulation-fragmentation models, where both the coagulation and fragmentation components of the collision operator are nonlinear. The models come from $3-$wave kinetic equations, a pivotal framework in wave turbulence theory. Despite the importance of wave turbulence theory in physics and mechanics, there have been very few numerical schemes for $3-$wave kinetic equations, in which no ad-hoc additional assumptions are imposed on the evolution of the solutions, and the current manuscript provides one of the first of such schemes. To the best of our knowledge, this also is the first numerical scheme capable of accurately capturing the long-term asymptotic behavior of solutions to a fully nonlinear coagulation-fragmentation model that includes both forward and backward energy cascades. The scheme is implemented on some test problems, demonstrating strong alignment with theoretical predictions of energy cascade rates. We further introduce a weighted Finite Volume variant to ensure energy conservation across varying degrees of kernel homogeneity. Convergence and first-order consistency are established through theoretical analysis and verified by experimental convergence orders in test cases.