On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation. (English) Zbl 1483.76044
Summary: Spectral methods, thanks to the high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the Boltzmann collision operator. On the other hand, the loss of some local invariants leads to the wrong long time behavior. A way to overcome this drawback, without sacrificing spectral accuracy, has been proposed recently with the construction of equilibrium preserving spectral methods. Despite the ability to capture the steady state with arbitrary accuracy, the theoretical properties of the method have never been studied in detail. In this paper, using the perturbation argument developed by F. Filbet and C. Mouhot [Trans. Am. Math. Soc. 363, No. 4, 1947–1980 (2011; Zbl 1213.82069)] for the homogeneous Boltzmann equation, we prove stability, convergence and spectrally accurate long time behavior of the equilibrium preserving approach.
MSC:
| 76M22 | Spectral methods applied to problems in fluid mechanics |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
Boltzmann equation; Fourier-Galerkin spectral method; steady-state preservation; micro-macro decomposition; local Maxwellian; stability; convergenceCitations:
Zbl 1213.82069Software:
FV_HipoDiffReferences:
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