Convergence and Error Estimates for the Lagrangian-Based Conservative Spectral Method for Boltzmann Equations

We develop error estimates for the semidiscrete conservative spectral method for the approximation of the elastic and inelastic space homogeneous Boltzmann equation introduced by Gamba and Tharkabhushanam in [ J. Comput. Phys., 228 (2009), pp. 2012--2036]. In addition we study the long time convergence of such semidiscrete solution to the equilibrium Maxwellian distribution that conserves the mass, momentum, and energy associated with the initial data. The numerical method is based on the Fourier transform of the collisional operator and a Lagrangian optimization correction that enforces the collision invariants, namely, conservation of mass, momentum, and energy in the elastic case, and just mass and momentum in the inelastic one. We present a detailed semidiscrete analysis on convergence of the proposed numerical method which includes the L¹-L² theory for the scheme. This analysis allows us to present, additionally, convergence in Sobolev spaces and convergence to equilibrium for the numerical approximation. The results of this work answer a long standing open problem posed by Cercignani, Illner, and Pulvirenti in [ Mathematical Theory of Dilute Gases, Springer, New York, 1994, Chapter 12] about finding error estimates for a numerical scheme associated with the Boltzmann equation, as well as showing the semidiscrete numerical solution converges to the equilibrium Maxwellian distribution associated with the initial value problem.