{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:58:34Z","timestamp":1776805114506,"version":"3.51.2"},"reference-count":34,"publisher":"American Mathematical Society (AMS)","issue":"319","license":[{"start":{"date-parts":[[2019,12,7]],"date-time":"2019-12-07T00:00:00Z","timestamp":1575676800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n 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 \n \n \n \n N<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n N^{-q}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the number of modes and\n \n \n \n \n q<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n q>0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n depends on the regularity of the solution. However, in the multiscale setting, the error estimate can be expressed in terms of the scaling parameter\n \n \n \n \n \u03f5\n \n <\/mml:mi>\n \\epsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , 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\n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n \n \u03f5\n \n <\/mml:mi>\n \n N<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathcal {O}(\\epsilon ^{N+1})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . More surprisingly, the coefficients of the expansion satisfy super convergence properties. In particular, the error of the\n \n \n \n \n \n \u2113\n \n <\/mml:mi>\n th<\/mml:mtext>\n <\/mml:mrow>\n \\ell \\text {th}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n coefficient of the expansion scales like\n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n \n \u03f5\n \n <\/mml:mi>\n \n 2<\/mml:mn>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathcal {O}(\\epsilon ^{2N})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n when\n \n \n \n \n \n \u2113\n \n <\/mml:mi>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\ell =0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n \n \u03f5\n \n <\/mml:mi>\n \n 2<\/mml:mn>\n N<\/mml:mi>\n +<\/mml:mo>\n 2<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n \n \u2113\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathcal {O}(\\epsilon ^{2N+2-\\ell })<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for all\n \n \n \n \n 1<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n \n \u2113\n \n <\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n 1\\leq \\ell \\leq N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\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 \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the time\n \n \n \n t<\/mml:mi>\n t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and the initial condition. We investigate specifically the dependence on\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , in order to assess whether increasing\n \n \n \n N<\/mml:mi>\n N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n actually yields an additional factor of\n \n \n \n \n \u03f5\n \n <\/mml:mi>\n \\epsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in the error. Numerical tests will also be presented to support the theoretical results.\n <\/p>","DOI":"10.1090\/mcom\/3399","type":"journal-article","created":{"date-parts":[[2018,9,19]],"date-time":"2018-09-19T10:16:00Z","timestamp":1537352160000},"page":"2257-2293","source":"Crossref","is-referenced-by-count":3,"title":["Multiscale convergence properties for spectral approximations of a model kinetic equation"],"prefix":"10.1090","volume":"88","author":[{"given":"Zheng","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cory","family":"Hauck","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2018,12,7]]},"reference":[{"key":"1","unstructured":"Multiprecision Computing Toolbox for MATLAB 4.3.3.12177, Advanpix LLC., Yokohama, Japan."},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"617","DOI":"10.2307\/1999099","article-title":"Diffusion approximation and computation of the critical size","volume":"284","author":"Bardos, C.","year":"1984","journal-title":"Trans. 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