Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 58, Issue 2, Pages 1164-1194Publisher
SIAM PUBLICATIONS
DOI: 10.1137/19M1270884
Keywords
convergence analysis; Grad's expansion; initial boundary value problem; kinetic equations
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Funding
- Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [320021702/GRK2326]
- BadenWuerttemberg Foundation via the project Numerical Methods for Multi-phase Flows with Strongly Varying Mach Numbers
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In [Commun. Pure. Appl. Math., 2 (1949), pp. 331-407], Grad proposed a Hermite series expansion for approximating solutions to kinetic equations that have an unbounded velocity space. However, for initial boundary value problems, poorly imposed boundary conditions lead to instabilities in Grad's Hermite expansion, which could result in nonconverging solutions. For linear kinetic equations, a method for posing stable boundary conditions was recently proposed for (formally) arbitrary order Hermite approximations. In the present work, we study L-2-convergence of these stable Hermite approximations and prove explicit convergence rates under suitable regularity assumptions on the exact solution. We confirm the presented convergence rates through numerical experiments involving the linearized BGK equation of rarefied gas dynamics.
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