A fast Fourier spectral method for wave kinetic equation

Abstract

The central object in wave turbulence theory is the wave kinetic equation (WKE), which is an evolution equation for wave action density and acts as the wave analog of the Boltzmann kinetic equations for particle interactions. Despite recent exciting progress in the theoretical aspects of the WKE, numerical developments have lagged behind. In this paper, we introduce a fast Fourier spectral method for solving the WKE. The key idea lies in reformulating the high-dimensional nonlinear wave kinetic operator as a spherical integral, analogous to the classical Boltzmann collision operator. The conservation of mass and momentum leads to a double convolution structure in Fourier space, which can be efficiently handled by the fast Fourier transform (FFT), reducing the computational cost from $O(N^{3d})$ to $O(M{N}^d\log N)$ with $N$-frequency nodes and $M\ll N^{2d-1}$ in $d$ dimensions. We demonstrate the accuracy and efficiency of the proposed method through several numerical tests in both 2D and 3D, revealing and conjecturing some interesting and unique features of this equation.

Updated version to appear in Journal of Computational Physics

References in corpus (11)

• Fast algorithms for computing the Boltzmann collision operator
• Solving the Boltzmann equation in N log N
• Weakly nonlinear Schrödinger equation with random initial data
• On the derivation of the wave kinetic equation for NLS
• Full derivation of the wave kinetic equation
• A fast Fourier spectral method for the homogeneous Boltzmann equation with non-cutoff collision kernels
• Direct verification of the kinetic description of wave turbulence for finite-size systems dominated by interactions among groups of 6 waves
• A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation
• A fast spectral method for the Boltzmann collision operator with general collision kernels
• A Deep Learning Approximation of Non-Stationary Solutions to Wave Kinetic Equations
• $L^2$-stability near equilibrium for the $4$ waves kinetic equation