[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ skip to main content
research-article

Perfectly matched layers for the Boltzmann equation: : Stability and sensitivity analysis

Published: 18 July 2024 Publication History

Abstract

We study the stability and sensitivity of an absorbing layer for the Boltzmann equation by examining the Bhatnagar–Gross–Krook (BGK) approximation and using the perfectly matched layer (PML) technique. To ensure stability, we discard some parameters in the model and calculate the total sensitivity indices of the remaining parameters using the ANOVA expansion of multivariate functions. We conduct extensive numerical experiments on two test cases to study stability and compute the total sensitivity indices, which allow us to identify the essential parameters of the model.

Highlights

Some parameters need to be set to zero in order to ensure the stability of the BGK model with the PML.
The most crucial parameters are the PML exponent and the PML thickness.
The values of the total sensitivity indices do not significantly depend on the choice of error functional.
Extensive numerical experiments confirm the theoretical analysis.

References

[1]
P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (1954) 511–525. https://link.aps.org/doi/10.1103/PhysRev.94.511.
[2]
J. Tölke, M. Krafczyk, M. Schulz, E. Rank, Discretization of the Boltzmann equation in velocity space using a Galerkin approach, Comput. Phys. Commun. 129 (2000) 91–99,.
[3]
J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994) 185–200,.
[4]
T. Hagstrom, A new construction of perfectly matched layers for hyperbolic systems with applications to the linearized Euler equations, in: Mathematical and Numerical Aspects of Wave Propagation—WAVES 2003, 2003, pp. 125–129,.
[5]
D. Appelö, T. Hagstrom, G. Kreiss, Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability, SIAM J. Appl. Math. 67 (2006) 1–23,.
[6]
Gao, Z.; Hesthaven, J.S.; Warburton, T. (2011): Efficient absorbing layers for weakly compressible flows. http://infoscience.epfl.ch/record/190661 submitted for publication, J. Sci. Comput. (Springer, ISSN 0885-7474).
[7]
L.C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, second ed., American Mathematical Society, Providence, RI, 2010,.
[8]
S. Chapman, T.G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge Math. Lib., third ed., Cambridge University Press, Cambridge, 1995, An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, in co-operation with D. Burnett, with a foreword by Carlo Cercignani.
[9]
H. Grad, On the kinetic theory of rarefied gases, Commun. Pure Appl. Math. 2 (1949) 331–407,.
[10]
A. Karakus, N. Chalmers, J.S. Hesthaven, T. Warburton, Discontinuous Galerkin discretizations of the Boltzmann–BGK equations for nearly incompressible flows: semi-analytic time stepping and absorbing boundary layers, J. Comput. Phys. 390 (2019) 175–202,.
[11]
D.C. Hernquist, Smoothly symmetrizable hyperbolic systems of partial differential equations, Math. Scand. 61 (1987) 262–275,.
[12]
S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89–112,.
[13]
M. Sutti, Analysis and optimization of perfectly matched layers for the Boltzmann equation, Master's thesis, EPFL, Computational Mathematics and Simulation Science 2015, http://refhub.elsevier.com/S0021-9991(19)30233-5/bib73757474695F616E616C797369735F32303135s1.
[14]
L. Rezzolla, Numerical Methods for the Solution of Hyperbolic Partial Differential Equations, 2005.
[15]
R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007,.
[16]
B. Gustafsson, H.-O. Kreiss, J. Oliger, Time-Dependent Problems and Difference Methods, Pure and Appl. Math. (Hoboken), second ed., John Wiley & Sons, Inc., Hoboken, NJ, 2013,.
[17]
E. Frank, On the zeros of polynomials with complex coefficients, Bull. Am. Math. Soc. 52 (1946) 144–157,.
[18]
M. Marden, Geometry of Polynomials, Math. Surveys, vol. 3, American Mathematical Society, Providence, RI, 1966,.
[19]
T. Andres, Sampling methods and sensitivity analysis for large parameter sets, J. Stat. Comput. Simul. 57 (1997) 77–110,.
[20]
A. Saltelli, K. Chan, E.M. Scott (Eds.), Sensitivity Analysis, in: Wiley Ser. Probab. Stat., Wiley, Chichester, 2000.
[21]
Y. Cao, Z. Chen, M. Gunzburger, ANOVA expansions and efficient sampling methods for parameter dependent nonlinear PDEs, Int. J. Numer. Anal. Model. 6 (2009) 256–273.
[22]
Z. Gao, J.S. Hesthaven, Efficient solution of ordinary differential equations with high-dimensional parametrized uncertainty, Commun. Comput. Phys. 10 (2011) 253–278,.
[23]
X. Wang, K.-T. Fang, The effective dimension and quasi-Monte Carlo integration, J. Complex. 19 (2003) 101–124,.
[24]
F.Q. Hu, X. Li, D. Lin, Absorbing boundary conditions for nonlinear Euler and Navier–Stokes equations based on the perfectly matched layer technique, J. Comput. Phys. 227 (2008) 4398–4424,.

Recommendations

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Journal of Computational Physics
Journal of Computational Physics  Volume 509, Issue C
Jul 2024
579 pages

Publisher

Academic Press Professional, Inc.

United States

Publication History

Published: 18 July 2024

Author Tags

  1. BGK model
  2. Perfectly matched layer
  3. Differential operators
  4. Stability analysis
  5. ANOVA expansion
  6. Total sensitivity index

Qualifiers

  • Research-article

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • 0
    Total Citations
  • 0
    Total Downloads
  • Downloads (Last 12 months)0
  • Downloads (Last 6 weeks)0
Reflects downloads up to 24 Dec 2024

Other Metrics

Citations

View Options

View options

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media