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This paper develops a new numerical method for solving gas dynamics equations based on the Boltzmann equation, using a compactly supported "hat function" instead of the traditional Maxwellian distribution. The hat function's limited velocity space support significantly simplifies mathematical proofs, allowing the authors to directly demonstrate that their scheme maintains physically required positive values. The method achieves second-order accuracy while preserving positivity and accurately captures contact discontinuities in fluid flows, as validated through standard one- and two-dimensional test problems.
Why it matters
Improved numerical schemes for gas dynamics have applications in aerospace engineering, weather prediction, and astrophysics simulations. The simplified mathematical framework could make these computational methods more reliable and easier to implement in scientific computing software, while the positivity-preserving property ensures physically realistic solutions.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: The suitable moments of the Maxwell velocity distribution function recover the conserved variable and flux vectors of the Euler equations of gas dynamics. However, the Maxwellian has infinite support in velocity space, which poses challenges for the numerical analysis of kinetic schemes based on it. Although Maxwellian is a natural choice, it is not the only distribution function that preserves these key moments. In his seminal paper, Perthame (1990) introduced a compactly supported hat function as an alternative. This paper presents the formulation and analysis of a contact discontinuity capturing Boltzmann scheme based on peculiar velocity, constructed using the hat function. Owing to the compact support of the hat function in velocity space, the numerical analysis of the proposed scheme is considerably simplified, allowing for a direct proof of its positivity. The scheme, extended to second-order accuracy without compromising the positivity-preserving property of its first-order counterpart, is tested on several benchmark inviscid flow problems in one- and two-dimensions.
Source: A hat function based positive contact discontinuity capturing Boltzmann scheme