Abstract:

This work develops and rigorously analyzes a unified Fourier spectral method for two fundamental collisional kinetic models: the homogeneous Landau equation and the spatially homogeneous cutoff Boltzmann equation. For both equations, we construct numerical solutions by truncating the velocity domain to a bounded box ( D_L ) and retaining ( N ) Fourier modes. A central contribution is the derivation of explicit error estimates for the proposed schemes, bounding the difference between the numerical and exact solutions in terms of the truncation parameters ( D_L ) and ( N ) for Maxwellian and hard potentials. Comprehensive numerical simulations are presented for both equations, which confirm the predicted convergence rates and demonstrate the methods' ability to capture essential solution dynamics, including the relaxation to equilibrium. Our results establish the first mathematically rigorous justification for Fourier spectral methods applied to the Landau equation while extending and complementing recent convergence analyses for the Boltzmann equation. Together, this work provides a complete, validated computational framework—from theoretical error analysis to practical implementation—for the reliable simulation of these foundational kinetic equations.