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These algorithms are implemented for the solution of the Boltzmann equation in two and three dimensions, first for homogeneous solutions, then for general nonhomogeneous solutions. The results are compared to explicit solutions, when available, and to Monte Carlo methods. In particular, the computational cost and accuracy are compared to those of Monte Carlo methods as well as to those of previous spectral methods. Finally, for inhomogeneous solutions, we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium, which was suggested in the work [L. Desvillettes and C. Villani, Invent. Math., 159 (2005), pp. 245-316].<\/jats:p>","DOI":"10.1137\/050625175","type":"journal-article","created":{"date-parts":[[2006,8,24]],"date-time":"2006-08-24T15:30:00Z","timestamp":1156433400000},"page":"1029-1053","source":"Crossref","is-referenced-by-count":90,"title":["Solving the Boltzmann Equation in\n                    <i>N<\/i>\n                    log\n                    <sub>2<\/sub>\n                    <i>N<\/i>"],"prefix":"10.1137","volume":"28","author":[{"given":"Francis","family":"Filbet","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cl\u00e9ment","family":"Mouhot","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lorenzo","family":"Pareschi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,8,4]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0914010"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/mma.1670080114"},{"key":"R3","volume-title":"Molecular gas dynamics and the direct simulation of gas flows","author":"Bird G.","year":"1995"},{"key":"R4","first-page":"1296","volume":"225","author":"Bobyl\u00ebv A.","year":"1975","journal-title":"Dokl. 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