Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations


Authors: Yingda Cheng, Irene M. Gamba and Jennifer Proft
Journal: Math. Comp. 81 (2012), 153-190
MSC (2010): Primary 65M60, 76P05, 74S05
DOI: https://doi.org/10.1090/S0025-5718-2011-02504-4
Published electronically: June 15, 2011
MathSciNet review: 2833491
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We develop a high-order positivity-preserving discontinuous Galerkin (DG) scheme for linear Vlasov-Boltzmann transport equations (Vlasov-BTE) under the action of quadratically confined electrostatic potentials. The solutions of such BTEs are positive probability distribution functions and it is very challenging to have a mass-conservative, high-order accurate scheme that preserves positivity of the numerical solutions in high dimensions. Our work extends the maximum-principle-satisfying scheme for scalar conservation laws in a recent work by X. Zhang and C.-W. Shu to include the linear Boltzmann collision term. The DG schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy. A discussion of the standard semi-discrete DG schemes for the BTE are included as a foundation for the stability and error estimates for this new scheme. Numerical results of the relaxation models are provided to validate the method.


References [Enhancements On Off] (What's this?)

  • 1. V. R. A.M. Anile, G. Mascali.
    Recent developments in hydrodynamical modeling of semiconductors, mathematical problems in semiconductor physics.
    Lect. Notes Math., 1823:156.
  • 2. N. Ben Abdallah and M. L. Tayeb.
    Diffusion approximation for the one dimensional Boltzmann-Poisson system.
    Discrete Contin. Dyn. Syst. Ser. B, 4(4):1129-1142, 2004. MR 2082927 (2005f:35031)
  • 3. G. A. Bird.
    Molecular gas dynamics.
    Clarendon Press, Oxford, 1994.
  • 4. J. E. Broadwell.
    Study of the rarified shear flow by the discrete velocity method.
    J. Fluid Mech., 19:401-414, 1964. MR 0172651 (30:2870)
  • 5. M. J. Caceres, J. A. Carrillo, I. M. Gamba, A. Majorana, and C.-W. Shu.
    Deterministic kinetic solvers for charged particle transport in semiconductor devices.
    Transport Phenomena and Kinetic Theory Applications to Gases, Semiconductors, Photons, and Biological Systems, pages 151-171, 2007. MR 2334309 (2008h:82089)
  • 6. J. A. Carrillo, I. M. Gamba, A. Majorana, and C.-W. Shu.
    A WENO-solver for 1D non-stationary Boltzmann-Poisson system for semiconductor devices.
    J. Comput. Electron., 1:365-375, 2002.
  • 7. J. A. Carrillo, I. M. Gamba, A. Majorana, and C.-W. Shu.
    A direct solver for 2D non-stationary Boltzmann-Poisson systems for semiconductor devices: a MESFET simulation by WENO-Boltzmann schemes.
    J. Comput. Electron., 2:375-380, 2003.
  • 8. J. A. Carrillo, I. M. Gamba, A. Majorana, and C.-W. Shu.
    A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices. Performance and comparisons with Monte Carlo methods.
    J. Comput. Phys., 184:498-525, 2003. MR 1959405 (2003m:82087)
  • 9. J. A. Carrillo, I. M. Gamba, A. Majorana, and C.-W. Shu.
    2D semiconductor device simulations by WENO-Boltzmann schemes: efficiency, boundary conditions and comparison to Monte Carlo methods.
    J. Comput. Phys., 214:55-80, 2006. MR 2208670 (2006i:82094)
  • 10. Y. Cheng, I. M. Gamba, A. Majorana, and C.-W. Shu.
    Discontinuous Galerkin solver for the semiconductor Boltzmann equation.
    SISPAD 07, June 14-17, pages 257-260, 2007.
  • 11. Y. Cheng, I. M. Gamba, A. Majorana, and C.-W. Shu.
    Discontinuous Galerkin solver for Boltzmann-Poisson transients.
    J. Comput. Electron., 7:119-123, 2008.
  • 12. Y. Cheng, I. M. Gamba, A. Majorana, and C.-W. Shu.
    A Discontinuous Galerkin solver for Boltzmann-Poisson systems for semiconductor devices.
    Comput. Methods Appl. Mech. Eng., 198:3130-3150, 2009. MR 2567861 (2010j:82096)
  • 13. Y. Cheng, I. M. Gamba, A. Majorana, and C.-W. Shu.
    A discontinuous Galerkin solver for full-band Boltzmann-Poisson models.
    Proceeding of the IWCE13, pages 211-214, 2009.
  • 14. Y. Cheng, I. M. Gamba, A. Majorana, and C.-W. Shu.
    Performance of discontinuous Galerkin solver for semiconductor Boltzmann Equation.
    Proceedings of the IWCE14, pp. 227-230, 2010.
  • 15. P. Ciarlet.
    The finite element method for elliptic problems.
    North-Holland, Amsterdam, 1975. MR 0520174 (58:25001)
  • 16. B. Cockburn, B. Dong, and J. Guzmán.
    Optimal convergence of the original DG method for the transport-reaction equation on special meshes.
    SIAM J. Numer. Anal., 46:1250-1265, 2008. MR 2390992 (2009c:65299)
  • 17. B. Cockburn, B. Dong, J. Guzmán, and J. Qian.
    Optimal convergence of the original DG method on special meshes for transport variable velocity. SIAM J. Numer. Anal. 48:133-146, 2010. MR 2608362
  • 18. B. Cockburn, S. Hou, and C.-W. Shu.
    The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case.
    Math. Comput., 54:545-581, 1990. MR 1010597 (90k:65162)
  • 19. B. Cockburn, G. Kanschat, I. Perugia, and D. Schötzau.
    Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids.
    SIAM J. Numer. Anal., 39:264-285, 2001. MR 1860725 (2002g:65140)
  • 20. B. Cockburn, S. Y. Lin, and C.-W. Shu.
    TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems.
    J. Comput. Phys., 84:90-113, 1989. MR 1015355 (90k:65161)
  • 21. B. Cockburn and C.-W. Shu.
    TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework.
    Math. Comput., 52:411-435, 1989. MR 983311 (90k:65160)
  • 22. B. Cockburn and C.-W. Shu.
    The Runge-Kutta local projection p1-discontinuous Galerkin finite element method for scalar conservation laws.
    Math. Model. Num. Anal., 25:337-361, 1991. MR 1103092 (92e:65128)
  • 23. B. Cockburn and C.-W. Shu.
    The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems.
    J. Comput. Phys., 141:199-224, 1998. MR 1619652 (99c:65181)
  • 24. B. Cockburn and C.-W. Shu.
    Runge-Kutta discontinuous Galerkin methods for convection-dominated problems.
    J. Sci. Comput., 16:173-261, 2001. MR 1873283 (2002i:65099)
  • 25. M. G. Crandall and L. Tartar.
    Some relations between nonexpansive and order preserving mappings.
    Proc. Amer. Math. Soc., 78:385-390, 1980. MR 553381 (81a:47054)
  • 26. E. Fatemi and F. Odeh.
    Upwind finite difference solution of Boltzmann equation applied to electron transport in semiconductor devices.
    J. Comput. Phys., 108:209-217, 1993. MR 1242949 (94e:78023)
  • 27. I. Gamba and S. H. Tharkabhushaman.
    Spectral-Lagrangian based methods applied to computation of non-equilibrium statistical states.
    J. Comput. Phys., 228:2012-2036, 2009. MR 2500671 (2009m:82068)
  • 28. S. Gottlieb and C.-W. Shu.
    Total variation diminishing Runge-Kutta schemes.
    Math. Comput., 67:73-85, 1998. MR 1443118 (98c:65122)
  • 29. S. Gottlieb, C.-W. Shu, and E. Tadmor.
    Strong stability preserving high order time discretization methods.
    SIAM Review, 43:89-112, 2001. MR 1854647 (2002f:65132)
  • 30. Y. Guo.
    The Vlasov-Poisson-Boltzmann system near Maxwellians.
    Comm. Pure Appl. Math., 55(9):1104-1135, 2002. MR 1908664 (2003b:82050)
  • 31. Y. Guo.
    The Vlasov-Maxwell-Boltzmann system near Maxwellians.
    Invent. Math., 153(3):593-630, 2003. MR 2000470 (2004m:82123)
  • 32. B. Helffer and F. Nier.
    Hypoellliptic estimates and spectral theory for Fokker-Planck operators and Witten laplacians.
    Lecture Notes in Mathematics series, 1862, 2005. MR 2130405 (2006a:58039)
  • 33. F. Hérau.
    Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation.
    Asymptot. Anal., 46:349-359, 2006. MR 2215889 (2007b:35044)
  • 34. F. Hérau and F. Nier.
    Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with high degree potential.
    Arch. Ration. Mech. Anal., 171:151-218, 2004. MR 2034753 (2005f:82085)
  • 35. C. Johnson and J. Pitkäranta.
    An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation.
    Math. Comp., 46:1-26, 1986. MR 815828 (88b:65109)
  • 36. P. Lesaint and P.-A. Raviart.
    On a finite element method for solving the neutron transport equation.
    In Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), pages 89-123. Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974. MR 0658142 (58:31918)
  • 37. A. Majorana and R. Pidatella.
    A finite difference scheme solving the Boltzmann Poisson system for semiconductor devices.
    J. Comput. Phys., 174:649-668, 2001.
  • 38. C. Mouhot and L. Neumann.
    Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus.
    Nonlinearity, 4:969-998, 2006. MR 2214953 (2007c:82032)
  • 39. K. Nanbu.
    Direct simulation scheme derived from the Boltzmann equation. I. monocomponent gases.
    J. Phys. Soc. Jpn., 52:2042-2049, 1983.
  • 40. L. Pareschi and B. Perthame.
    A Fourier spectral method for homogeneous Boltzmann equations.
    Transport Theory Statist. Phys., 25:369-C383, 1996. MR 1407541 (97j:82133)
  • 41. T. E. Peterson.
    A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation.
    SIAM J. Numer. Anal., 28:133-140, 1991. MR 1083327 (91m:65250)
  • 42. M. Portelheiro and A. Tzvaras.
    Hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves.
    Trans. Amer. Math. Soc., 359:529-565, 2007. MR 2255185 (2007k:35323)
  • 43. W. Reed and T. Hill.
    Tiangular mesh methods for the neutron transport equation.
    Technical report, Los Alamos National Laboratory, Los Alamos, NM, 1973.
  • 44. L. Reggiani.
    Hot-electron transport in semiconductors, volume 58 of Topics in Applied Physics.
    Springer, Berlin, 1985.
  • 45. G. R. Richter.
    An optimal-order error estimate for the discontinuous Galerkin method.
    Math. Comp., 50:75-88, 1988. MR 917819 (88j:65197)
  • 46. C.-W. Shu and S. Osher.
    Efficient implementation of essentially non-oscillatory shock-capturing schemes.
    J. Comput. Phys., 77:439-471, 1988. MR 954915 (89g:65113)
  • 47. R. M. Strain and Y. Guo.
    Almost exponential decay near Maxwellian.
    Comm. Partial Differential Equations, 31(1-3):417-429, 2006. MR 2209761 (2006m:82042)
  • 48. K. Tomizawa.
    Numerical simulation of sub micron semiconductor devices.
    Artech House, Boston, 1993.
  • 49. C. Villani.
    A review of mathematical topics in collisional kinetic theory handbook of mathematical fluid dynamics.
    I:71-74, 2002. MR 1942465 (2003k:82087)
  • 50. C. Villani.
    Hypocoercive diffusion operators in Hörmander form.
    Mathematical Models and Methods in Applied Sciences, 2006. MR 2275692 (2007k:35057)
  • 51. X. Zhang and C.-W. Shu.
    A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws.
    SIAM Journal on Numerical Analysis, 48(2):772-795, 2010. MR 2670004
  • 52. X. Zhang and C.-W. Shu.
    On maximum-principle-satisfying high order schemes for scalar conservation laws.
    J. Comput. Phys., 229:3091-3120, 2010. MR 2601091 (2010k:65181)
  • 53. X. Zhang and C.-W. Shu.
    On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes.
    Journal of Computational Physics, 229:8918-8934, 2010. MR 2725380
  • 54. X. Zhang, Y. Xia, and C.-W. Shu.
    Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes.
    Journal of Scientific Computing.
    to appear
  • 55. X. Zhang, Y. Xing, and C.-W. Shu.
    Positivity preserving high order well balanced discontinuous Galerkin methods for the shallow water equations.
    Advances in Water Resources.
    33:1476-1493, 2010.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65M60, 76P05, 74S05

Retrieve articles in all journals with MSC (2010): 65M60, 76P05, 74S05


Additional Information

Yingda Cheng
Affiliation: Department of Mathematics and ICES, University of Texas at Austin, Austin, Texas 78712
Address at time of publication: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: ycheng@math.utexas.edu

Irene M. Gamba
Affiliation: Department of Mathematics and ICES, University of Texas at Austin, Austin, Texas 78712
Email: gamba@math.utexas.edu

Jennifer Proft
Affiliation: ICES, University of Texas at Austin, Austin, Texas 78712
Email: jennifer@ices.utexas.edu

DOI: https://doi.org/10.1090/S0025-5718-2011-02504-4
Keywords: Boltzmann transport equations, discontinuous Galerkin finite element methods, positivity-preserving schemes, stability, error estimates, relaxation models
Received by editor(s): July 23, 2010
Received by editor(s) in revised form: October 27, 2010
Published electronically: June 15, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society