Multiscale convergence properties for spectral approximations of a model kinetic equation
Zheng Chen, Cory D. Hauck · Mathematics of Computation · 2018
In this work, we prove rigorous convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form N − q N^{-q} , where N N is the number of modes and q > 0 q>0 depends on the regularity of the solution. However, in the multiscale setting, the error estimate can be expressed in terms of the scaling parameter ϵ \epsilon , which measures the ratio of the mean-free-path to the characteristic domain length. We show that, for isotropic initial conditions, the error in the spectral approximation is O ( ϵ N + 1 ) \mathcal {O}(\epsilon ^{N+1}) . More surprisingly, the coefficients of the expansion satisfy super convergence properties. In particular, the error of the ℓ th \ell \text {th} coefficient of the expansion scales like O ( ϵ 2 N ) \mathcal {O}(\epsilon ^{2N}) when ℓ = 0 \ell =0 and O ( ϵ 2 N + 2 − ℓ ) \mathcal {O}(\epsilon ^{2N+2-\ell }) for all 1 ≤