Numerical Schemes for 3-Wave Kinetic Equations: A Complete Treatment of the Collision Operator
<|reference_start|>Numerical Schemes for 3-Wave Kinetic Equations: A Complete Treatment of the Collision Operator: In our previous work, numerical schemes for a simplified version of 3-wave kinetic equations, in which only the simple forward-cascade terms of the collision operators are kept, have been successfully designed, especially to capture the long time dynamics of the equation given the multiple blow-up time phenomenon. In this second work in the series, we propose numerical treatments for the complete 3-wave kinetic equations, in which the complete, much more complicated collision operators are fully considered based on a novel conservative form of the equation. We then derive an implicit finite volume scheme to solve the equation. The new discretization uses an adaptive time-stepping method which allows for the simulations to be carried to very long times. Our computed solutions are compared with previously derived long-time asymptotic estimates for the decay rate of total energy of time-dependent solutions of 3-wave kinetic equations and found to be in excellent agreement.<|reference_end|>
@article{walton2024numerical,
    title={Numerical Schemes for 3-Wave Kinetic Equations: A Complete Treatment of
  the Collision Operator},
    author={Steven Walton and Minh-Binh Tran},
    journal={arXiv preprint arXiv:2402.17481},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.17481},
    primaryClass={math.NA cs.NA}
}