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Philipp Grohs; Ralf Hiptmair; Simon Pintarelli
Tensor-Product Discretization for the Spatially Inhomogeneous and Transient Boltzmann Equation in Two Dimensions
SMAI-Journal of computational mathematics, 3 (2017), p. 219-248
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Class. Math.: 76J20, 76H05, 76P05, 82C40, 82D05, 65Y05, 65M60

Abstract

We consider the spatially inhomogeneous and nonlinear Boltzmann equation for the variable hard spheres model. The distribution function is discretized by a tensor-product ansatz by combining Maxwellian modulated Laguerre polynomials in velocity with continuous, linear finite elements in the spatial domain. The advection problem in phase space is discretized through a Galerkin least squares technique and yields an implicit formulation in time. The discrete collision operator can be evaluated with an asymptotic effort of $\mathcal{O}(K^5)$, where $K$ is the number of velocity degrees of freedom in a single direction. Numerical results in 2D are presented for rarefied gases with different Mach and Knudsen numbers.

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