Volume 28, Issue 4
Shock and Boundary Structure Formation by Spectral-Lagrangian Methods for the Inhomogeneous Boltzmann Transport Equation

J. Comp. Math., 28 (2010), pp. 430-460.

Published online: 2010-08

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• Abstract

The numerical approximation of the Spectral-Lagrangian scheme developed by the authors in [30] for a wide range of homogeneous non-linear Boltzmann type equations is extended to the space inhomogeneous case and several shock problems are benchmark. Recognizing that the Boltzmann equation is an important tool in the analysis of formation of shock and boundary layer structures, we present the computational algorithm in Section 3.3 and perform a numerical study case in shock tube geometries well modeled in for $1D$ in $\textbf{x}$ times $3D$ in $\textbf{v}$ in Section 4. The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Aoki et al [2], where the wall temperature is suddenly increased or decreased, is also studied. We consider the problem of heat transfer between two parallel plates with diffusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefied state. Finally, the classical infinite shock tube problem that generates a non-moving shock wave is studied. The point worth noting in this example is that the flow in the final case turns from a supersonic flow to a subsonic flow across the shock.

• Keywords

Spectral Numerical Methods Lagrangian optimization FFT Boltzmann Transport Equation Conservative and non-conservative rarefied gas flows

• AMS Subject Headings

65T50 76P05 76M22 80A20 82B30 82B40 82B80

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COPYRIGHT: © Global Science Press

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@Article{JCM-28-430, author = {}, title = {Shock and Boundary Structure Formation by Spectral-Lagrangian Methods for the Inhomogeneous Boltzmann Transport Equation}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {4}, pages = {430--460}, abstract = {

The numerical approximation of the Spectral-Lagrangian scheme developed by the authors in [30] for a wide range of homogeneous non-linear Boltzmann type equations is extended to the space inhomogeneous case and several shock problems are benchmark. Recognizing that the Boltzmann equation is an important tool in the analysis of formation of shock and boundary layer structures, we present the computational algorithm in Section 3.3 and perform a numerical study case in shock tube geometries well modeled in for $1D$ in $\textbf{x}$ times $3D$ in $\textbf{v}$ in Section 4. The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Aoki et al [2], where the wall temperature is suddenly increased or decreased, is also studied. We consider the problem of heat transfer between two parallel plates with diffusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefied state. Finally, the classical infinite shock tube problem that generates a non-moving shock wave is studied. The point worth noting in this example is that the flow in the final case turns from a supersonic flow to a subsonic flow across the shock.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1003-m0011}, url = {http://global-sci.org/intro/article_detail/jcm/8531.html} }
TY - JOUR T1 - Shock and Boundary Structure Formation by Spectral-Lagrangian Methods for the Inhomogeneous Boltzmann Transport Equation JO - Journal of Computational Mathematics VL - 4 SP - 430 EP - 460 PY - 2010 DA - 2010/08 SN - 28 DO - http://doi.org/10.4208/jcm.1003-m0011 UR - https://global-sci.org/intro/article_detail/jcm/8531.html KW - Spectral Numerical Methods KW - Lagrangian optimization KW - FFT KW - Boltzmann Transport Equation KW - Conservative and non-conservative rarefied gas flows AB -

The numerical approximation of the Spectral-Lagrangian scheme developed by the authors in [30] for a wide range of homogeneous non-linear Boltzmann type equations is extended to the space inhomogeneous case and several shock problems are benchmark. Recognizing that the Boltzmann equation is an important tool in the analysis of formation of shock and boundary layer structures, we present the computational algorithm in Section 3.3 and perform a numerical study case in shock tube geometries well modeled in for $1D$ in $\textbf{x}$ times $3D$ in $\textbf{v}$ in Section 4. The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Aoki et al [2], where the wall temperature is suddenly increased or decreased, is also studied. We consider the problem of heat transfer between two parallel plates with diffusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefied state. Finally, the classical infinite shock tube problem that generates a non-moving shock wave is studied. The point worth noting in this example is that the flow in the final case turns from a supersonic flow to a subsonic flow across the shock.

Irene M. Gamba & Sri Harsha Tharkabhushanam. (2019). Shock and Boundary Structure Formation by Spectral-Lagrangian Methods for the Inhomogeneous Boltzmann Transport Equation. Journal of Computational Mathematics. 28 (4). 430-460. doi:10.4208/jcm.1003-m0011
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