Algorithm for Solving the Four-Wave Kinetic Equation in Problems of Wave Turbulence

We propose the method for numerical solution of four-wave kinetic equations that arise in the wave turbulence (weak turbulence) theory when describing a homogeneous isotropic interaction of waves. To calculate the collision integral in the right-hand side of equation, the cubature formulas of high rate of convergence are developed, which allow for adaptation of the algorithm to the singularities of the solutions and of the integral kernels. The convergence tests in the problems of integration arising from real applications are done. To take into account the multi-scale nature of turbulence problems in our algorithm, rational approximations of the solutions and a new time marching scheme are implemented and tested. The efficiency of the developed algorithm is demonstrated by modelling the inverse cascade of Bose gas particles during the formation of a Bose–Einstein condensate.