Multiscale convergence properties for spectral approximations of a model kinetic equation
For researchers in kinetic theory and multiscale modeling, this work provides theoretical error estimates that improve understanding of spectral methods in the small-ε regime.
This paper proves rigorous convergence properties for a spectral approximation of a kinetic equation, showing that the error scales as O(ε^{N+1}) for isotropic initial conditions and that low-order coefficients exhibit super convergence up to O(ε^{2N}).
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 $N^{-q}$, where $N$ is the number of modes and $q>0$ depends on the regularity of the solution. However, in the multiscale setting, the error estimate can be expressed in terms of the scaling parameter $ε$, 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 $\mathcal{O}(ε^{N+1})$. More surprisingly, the coefficients of the expansion satisfy super convergence properties. In particular, the error of the $\ell^{th}$ coefficient of the expansion scales like $\mathcal{O}(ε^{2N})$ when $\ell =0$ and $\mathcal{O}(ε^{2N+2-\ell})$ for all $1\leq \ell \leq N$. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system. All the above estimates involve constants depending on $N$, the time $t$, and the initial condition. We investigate specifically the dependence on $N$, in order to assess whether increasing $N$ actually yields an additional factor of $ε$ in the error. Numerical tests will also be presented to support the theoretical results.