Analysis of a numerical scheme for 3-wave kinetic equations

Abstract

Several numerical schemes for 3-wave kinetic equations have been proposed in recent work and shown to be accurate and computationally efficient [8,33,34,35]. However, their rigorous numerical analysis remains open. This paper aims to close this gap. We establish a comprehensive well-posedness and qualitative theory for the discrete equation arising from those schemes. We prove global existence, uniqueness, and Lipschitz stability of nonnegative classical solutions in $\ell^1(\mathbb{N})$, together with uniform bounds and decay of moments. We further show exponential energy decay and a sharp creation and propagation of positivity characterized by the arithmetic structure of the initial support. Finally, we obtain the propagation and instantaneous creation of polynomial, Mittag-Leffler, and exponential moments, providing quantitative control of high energy tails. We validate the theoretical findings by numerical results.