Analysis of a numerical scheme for 3-wave kinetic equations

TL;DR

This work provides a rigorous analysis of a discrete finite-volume scheme for isotropic 3-wave kinetic equations, establishing global existence, uniqueness, and Lipschitz stability of nonnegative solutions in $ell1(N)$ with uniform moment bounds and exponential energy decay. It introduces a forward-invariant viable set and a tangency framework to obtain well-posedness, and proves sharp positivity creation/propagation on the gcd lattice of the initial support along with propagation and creation results for polynomial, Mittag-Leffler, and exponential moments. The results are complemented by numerical tests that corroborate positivity, moment propagation, and tail behavior, thereby furnishing a solid theoretical foundation for the discretized 3-wave kinetics and its computational use.

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.

Figures (10)

• Figure 1: Initial data $f_0$.
• Figure 2: Solution at $T=1$ (left) and at $T=10$ (right).
• Figure 3: $m_1\left\langle f(t)\right\rangle$ (left) and $m_2\left\langle f(t)\right\rangle$ (right) on $[0,1]$.
• Figure 4: $m_3\left\langle f(t)\right\rangle$ (left) and $m_4\left\langle f(t)\right\rangle$ (right) on $[0,1]$.
• Figure 5: Mittag-Leffler moments $\mathcal{E}_a^\infty(\lambda)\left\langle f(t)\right\rangle$ for $a=1,\lambda=0.1$ (left) and for $a=1,\lambda=1$ (right) on $[0,1]$.

Theorems & Definitions (44)

• Definition 2.1
• Remark 2.2
• Theorem 2.4: Existence and Uniqueness
• Theorem 2.5: Lipschitz Continuity
• Theorem 2.6
• Remark 2.7
• Theorem 2.8: Creation and Propagation of Positivity
• Corollary 2.9: Propagation of Polynomial Moments
• proof
• Theorem 2.10: Creation of Polynomial Moments