A fast Fourier spectral method for wave kinetic equation

TL;DR

This work addresses the numerical solution of the wave kinetic equation (WKE), a high-dimensional nonlinear evolution equation arising in wave turbulence. The authors extend the Fourier spectral method to the WKE by reformulating the nonlinear operator as a sphere-based Boltzmann-type integral, revealing a double-convolution structure in Fourier space that can be accelerated with the fast Fourier transform (FFT). A fast algorithm is developed via a low-rank decomposition of the collision weight, reducing the computational cost from $O(N^{3d})$ to $O(M N^d \log N)$ and enabling on-the-fly evaluation without precomputation. The method is validated in 2D and 3D through stationary RJ tests and time-evolution scenarios for isotropic and non-isotropic initial data, demonstrating accurate equilibria, mass/energy conservation, and complex dynamical features such as blow-up and concentration shifts. The approach offers a scalable, deterministic framework for numerically exploring WKE dynamics and can be extended to other dispersion relations and higher dimensions.

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 classical Boltzmann collision operator. The conservation of mass and momentum leads to a double convolution structure in Fourier space, which can be efficiently handled using the fast Fourier transform. We demonstrate the accuracy and efficiency of the proposed method through several numerical tests in both 2D and 3D, revealing some interesting and unique features of this equation.

Figures (6)

• Figure 1: Stationary solution in 2D with $S = 5, N_r = N = 128, N_s = N_{sig} = 12$.
• Figure 2: Top-to-bottom view of the time evolution of the solution profile for the isotropic initial condition \ref{['initialdelta']} with $N_r = N = 64, N_s = N_{sig} = 12$, $S=0.33$ and $\Delta t = 0.1$.
• Figure 3: Discontinous initial condition $f^0_{\text{dis}}$ in 2D for $N = N_r=64, N_s = N_{sig} = 12$, $S = 3$.
• Figure 4: Evolution of mass and energy in 2D for $N = N_r, N_s = N_{sig} = 12$ and $\Delta t = 0.1$.
• Figure 5: Time evolution of the solution profile for the non-isotropic initial condition with $N = N_r = 64, N_s = N_{sig} = 8$, $S=3$ and $\Delta t = 1$.