Analysis of a numerical scheme for 3-wave kinetic equations

TL;DR

This paper provides a rigorous mathematical analysis of numerical schemes for 3-wave kinetic equations, establishing well-posedness, stability, decay properties, and moment creation, supported by numerical validation.

Contribution

It offers the first comprehensive theoretical framework for these schemes, including existence, uniqueness, stability, and moment behavior analysis.

Findings

Proved global existence and uniqueness of solutions.

Established exponential energy decay and positivity propagation.

Demonstrated creation of high-energy moments and validated with numerical results.

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.