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Mathematics of Computation

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Multiscale convergence properties for spectral approximations of a model kinetic equation

Authors: Zheng Chen and Cory D. Hauck
Journal: Math. Comp. 88 (2019), 2257-2293
MSC (2010): Primary 65B10, 65G99, 65M15, 65M22, 65M70, 65Z05
Published electronically: December 7, 2018
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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$$ \text {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.

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Additional Information

Zheng Chen
Affiliation: Department of Mathematics, University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, Massachusetts 02747

Cory D. Hauck
Affiliation: Computational and Applied Mathematics Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831

Keywords: Kinetic equation, multiscale, super convergence, spectral method, diffusion approximation
Received by editor(s): April 20, 2018
Received by editor(s) in revised form: July 17, 2018, and September 8, 2018
Published electronically: December 7, 2018
Additional Notes: This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research.
This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (