A Numerical Scheme for Wave Turbulence: 3-Wave Kinetic Equations

TL;DR

This paper presents a novel finite volume numerical scheme for isotropic 3-wave kinetic equations that accurately captures the long-term energy cascade behavior, aligning well with theoretical predictions.

Contribution

It introduces the first numerical scheme capable of simulating the long-time asymptotic behavior of isotropic 3-wave kinetic equations, utilizing a new identity to simplify collision computations.

Findings

Numerical solutions verify the energy cascade phenomenon.

Energy cascade rates match theoretical predictions.

The scheme effectively captures long-term asymptotic behavior.

Abstract

We introduce a finite volume scheme to solve a special case of isotropic 3-wave kinetic equations. We test our numerical solution against theoretical results concerning the long time behavior of the energy and observe that our solutions verify the energy cascade phenomenon. To our knowledge, this is the first numerical scheme that can capture the long time asymptotic behavior of solutions to those isotropic 3-wave kinetic equations, where the energy cascade can be observed. Our numerical energy cascade rates are in good agreement with previously obtained theoretical results. The finite volume scheme given here relies on a new identity, allowing one to reduce the number of terms needed in the collision operators.