Fast spectral methods for Boltzmann and Landau integral operators of gas and plasma kinetic theory (Q2718086)

The nonlinear partial integro-differential equation NEWLINE\[NEWLINE \frac{\partial f}{\partial t}+v\cdot\nabla_xf=Q(f,f), \quad f=f(x,v,t),\quad x,v\in R^3 NEWLINE\]NEWLINE is considered. Using a standard splitting algorithm the homogeneous equation of the form NEWLINE\[NEWLINE \frac{\partial f}{\partial t}=Q(f,f), \quad f=f(v),\quad v\in R^3 NEWLINE\]NEWLINE is obtained. A new version of the Fourier-Galerkin method for this equation based on a Fourier spectral approximation of the bilinear quadratic integral operator \(Q(f,f)\) is presented. The consistency and the spectral accuracy are estimated. The computational complexity of the scheme is \(O(n^2)\) for the Boltzmann integral and \(O(n\log_2n)\) for the Landau collisional integral. Numerical results are given in 2D and 3D spaces: error, convergence rate and CPU time (Intel Pentium 266, Fortran 77).