A hat function based positive contact discontinuity capturing Boltzmann scheme
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.
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BibTeX
@misc{singh2026function,
title = {{A hat function based positive contact discontinuity capturing Boltzmann scheme}},
author = {Balwinder Singh and S. V. Raghurama Rao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10419},
eprint = {2610.10419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10419},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A hat function based positive contact discontinuity capturing Boltzmann scheme
AU - Balwinder Singh
AU - S. V. Raghurama Rao
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10419
UR - https://arxiv.org/abs/2610.10419
AB - 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.
ER -