A fast Fourier spectral method for wave kinetic equation
This work provides a computationally efficient numerical method for solving the wave kinetic equation, which is important for wave turbulence theory but has lagged in numerical development.
The paper introduces a fast Fourier spectral method for solving the wave kinetic equation, reducing computational cost from O(N^{3d}) to O(M N^d log N) with M << N^{2d-1} in d dimensions, and demonstrates accuracy and efficiency through numerical tests in 2D and 3D.
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.