Multiscale convergence properties for spectral approximations of a model kinetic equation

TL;DR

This paper proves multiscale convergence and super convergence properties for spectral approximations of a kinetic equation, showing that errors decrease rapidly with the number of modes and the scale parameter.

Contribution

It establishes rigorous error estimates and super convergence results for spectral methods applied to a kinetic equation in a multiscale setting, with detailed dependence on the number of modes and scale.

Findings

Error in spectral approximation scales as (psilon^{N+1}) for isotropic initial conditions.

Coefficients of the expansion exhibit super convergence, with errors scaling as (psilon^{2N}) and (psilon^{2N+2-\u001ell}).

Numerical tests support the theoretical convergence and super convergence results.

Abstract

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 $\epsilon$, 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 $\mathcal{O}(\epsilon^{N+1})$. 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 $\mathcal{O}(\epsilon^{2N})$ when $\ell =0$ and $\mathcal{O}(\epsilon^{2N+2-\ell})$ 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 $\epsilon$ in the error. Numerical tests will also be presented to support the theoretical results.