Wigner-Eckart Factorization of the Spectral Boltzmann Collision Operator
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Hiemstra, René R. Keßler, Torsten Abdelmalik, Michael R. A. |
| author_facet | Hiemstra, René R. Keßler, Torsten Abdelmalik, Michael R. A. |
| contents | We reduce the eight-dimensional weak form of the bilinear Boltzmann collision operator to a five-dimensional kinematic core by rigidly rotating the laboratory frame to align with the colliding pair and integrating over the $\mathrm{SO}(3)$ rotation group. This reduction yields an exact Wigner--Eckart factorization within a spectral Galerkin framework of associated Laguerre polynomials and spherical harmonics. The decomposition decouples the angular geometry from the scattering physics. The former, represented by Clebsch--Gordan coefficients, is evaluated exactly, while the latter is evaluated to machine precision by a spectrally convergent singular quadrature strategy. By explicitly zeroing specific entries, the macroscopic collision invariants are embedded without approximation. Cache-optimized contractions deliver up to a 37-fold single-core speedup and a 1000-fold memory reduction over standard dense Cartesian formulations. The approach is validated against analytical solutions for Maxwell molecules and infinite-order Chapman--Enskog viscosity coefficients for hard spheres. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_28475 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wigner-Eckart Factorization of the Spectral Boltzmann Collision Operator Hiemstra, René R. Keßler, Torsten Abdelmalik, Michael R. A. Numerical Analysis Computational Physics Fluid Dynamics 76P05, 82C40, 65D32, 65R20 J.2; G.1.4; G.1.9; G.4 We reduce the eight-dimensional weak form of the bilinear Boltzmann collision operator to a five-dimensional kinematic core by rigidly rotating the laboratory frame to align with the colliding pair and integrating over the $\mathrm{SO}(3)$ rotation group. This reduction yields an exact Wigner--Eckart factorization within a spectral Galerkin framework of associated Laguerre polynomials and spherical harmonics. The decomposition decouples the angular geometry from the scattering physics. The former, represented by Clebsch--Gordan coefficients, is evaluated exactly, while the latter is evaluated to machine precision by a spectrally convergent singular quadrature strategy. By explicitly zeroing specific entries, the macroscopic collision invariants are embedded without approximation. Cache-optimized contractions deliver up to a 37-fold single-core speedup and a 1000-fold memory reduction over standard dense Cartesian formulations. The approach is validated against analytical solutions for Maxwell molecules and infinite-order Chapman--Enskog viscosity coefficients for hard spheres. |
| title | Wigner-Eckart Factorization of the Spectral Boltzmann Collision Operator |
| topic | Numerical Analysis Computational Physics Fluid Dynamics 76P05, 82C40, 65D32, 65R20 J.2; G.1.4; G.1.9; G.4 |
| url | https://arxiv.org/abs/2605.28475 |