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

Cheng, Yingda; Gamba, Irene; Proft, Jennifer

doi:10.1090/S0025-5718-2011-02504-4

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations
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by Yingda Cheng, Irene M. Gamba and Jennifer Proft;

Math. Comp. 81 (2012), 153-190

DOI: https://doi.org/10.1090/S0025-5718-2011-02504-4

Published electronically: June 15, 2011

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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

V. R. A.M. Anile, G. Mascali. Recent developments in hydrodynamical modeling of semiconductors, mathematical problems in semiconductor physics. Lect. Notes Math., 1823:156.
N. Ben Abdallah and M. Lazhar Tayeb, Diffusion approximation for the one dimensional Boltzmann-Poisson system, Discrete Contin. Dyn. Syst. Ser. B 4 (2004), no. 4, 1129–1142. MR 2082927, DOI 10.3934/dcdsb.2004.4.1129
G. A. Bird. Molecular gas dynamics. Clarendon Press, Oxford, 1994.
James E. Broadwell, Study of rarefied shear flow by the discrete velocity method, J. Fluid Mech. 19 (1964), 401–414. MR 172651, DOI 10.1017/S0022112064000817
M. J. Cáceres, 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, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2007, pp. 151–171. MR 2334309, DOI 10.1007/978-0-8176-4554-0_{7}
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.
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.
José A. Carrillo, Irene M. Gamba, Armando Majorana, and Chi-Wang 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 (2003), no. 2, 498–525. MR 1959405, DOI 10.1016/S0021-9991(02)00032-3
José A. Carrillo, Irene M. Gamba, Armando Majorana, and Chi-Wang Shu, 2D semiconductor device simulations by WENO-Boltzmann schemes: efficiency, boundary conditions and comparison to Monte Carlo methods, J. Comput. Phys. 214 (2006), no. 1, 55–80. MR 2208670, DOI 10.1016/j.jcp.2005.09.005
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.
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.
Yingda Cheng, Irene M. Gamba, Armando Majorana, and Chi-Wang Shu, A discontinuous Galerkin solver for Boltzmann-Poisson systems in nano devices, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 37-40, 3130–3150. MR 2567861, DOI 10.1016/j.cma.2009.05.015
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.
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.
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
Bernardo Cockburn, Bo Dong, and Johnny Guzmán, Optimal convergence of the original DG method for the transport-reaction equation on special meshes, SIAM J. Numer. Anal. 46 (2008), no. 3, 1250–1265. MR 2390992, DOI 10.1137/060677215
Bernardo Cockburn, Bo Dong, Johnny Guzmán, and Jianliang Qian, Optimal convergence of the original DG method on special meshes for variable transport velocity, SIAM J. Numer. Anal. 48 (2010), no. 1, 133–146. MR 2608362, DOI 10.1137/080740805
Bernardo Cockburn, Suchung Hou, and Chi-Wang Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comp. 54 (1990), no. 190, 545–581. MR 1010597, DOI 10.1090/S0025-5718-1990-1010597-0
Bernardo Cockburn, Guido Kanschat, Ilaria Perugia, and Dominik Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal. 39 (2001), no. 1, 264–285. MR 1860725, DOI 10.1137/S0036142900371544
Bernardo Cockburn, San Yih Lin, and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems, J. Comput. Phys. 84 (1989), no. 1, 90–113. MR 1015355, DOI 10.1016/0021-9991(89)90183-6
Bernardo Cockburn and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp. 52 (1989), no. 186, 411–435. MR 983311, DOI 10.1090/S0025-5718-1989-0983311-4
Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 3, 337–361 (English, with French summary). MR 1103092, DOI 10.1051/m2an/1991250303371
Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, J. Comput. Phys. 141 (1998), no. 2, 199–224. MR 1619652, DOI 10.1006/jcph.1998.5892
Bernardo Cockburn and Chi-Wang Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (2001), no. 3, 173–261. MR 1873283, DOI 10.1023/A:1012873910884
Michael G. Crandall and Luc Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc. 78 (1980), no. 3, 385–390. MR 553381, DOI 10.1090/S0002-9939-1980-0553381-X
Emad Fatemi and Faroukh Odeh, Upwind finite difference solution of Boltzmann equation applied to electron transport in semiconductor devices, J. Comput. Phys. 108 (1993), no. 2, 209–217. MR 1242949, DOI 10.1006/jcph.1993.1176
Irene M. Gamba and Sri Harsha Tharkabhushanam, Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys. 228 (2009), no. 6, 2012–2036. MR 2500671, DOI 10.1016/j.jcp.2008.09.033
Sigal Gottlieb and Chi-Wang Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp. 67 (1998), no. 221, 73–85. MR 1443118, DOI 10.1090/S0025-5718-98-00913-2
Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), no. 1, 89–112. MR 1854647, DOI 10.1137/S003614450036757X
Yan Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math. 55 (2002), no. 9, 1104–1135. MR 1908664, DOI 10.1002/cpa.10040
Yan Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math. 153 (2003), no. 3, 593–630. MR 2000470, DOI 10.1007/s00222-003-0301-z
Bernard Helffer and Francis Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Mathematics, vol. 1862, Springer-Verlag, Berlin, 2005. MR 2130405, DOI 10.1007/b104762
Frédéric Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal. 46 (2006), no. 3-4, 349–359. MR 2215889
Frédéric Hérau and Francis Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151–218. MR 2034753, DOI 10.1007/s00205-003-0276-3
C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), no. 173, 1–26. MR 815828, DOI 10.1090/S0025-5718-1986-0815828-4
P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Academic Press, New York-London, 1974, pp. 89–123. MR 658142
A. Majorana and R. Pidatella. A finite difference scheme solving the Boltzmann Poisson system for semiconductor devices. J. Comput. Phys., 174:649–668, 2001.
Clément Mouhot and Lukas Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity 19 (2006), no. 4, 969–998. MR 2214953, DOI 10.1088/0951-7715/19/4/011
K. Nanbu. Direct simulation scheme derived from the Boltzmann equation. I. monocomponent gases. J. Phys. Soc. Jpn., 52:2042–2049, 1983.
Lorenzo Pareschi and Benoit Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), 1996, pp. 369–382. MR 1407541, DOI 10.1080/00411459608220707
Todd E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal. 28 (1991), no. 1, 133–140. MR 1083327, DOI 10.1137/0728006
Manuel Portilheiro and Athanasios E. Tzavaras, Hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves, Trans. Amer. Math. Soc. 359 (2007), no. 2, 529–565. MR 2255185, DOI 10.1090/S0002-9947-06-04268-1
W. Reed and T. Hill. Tiangular mesh methods for the neutron transport equation. Technical report, Los Alamos National Laboratory, Los Alamos, NM, 1973.
L. Reggiani. Hot-electron transport in semiconductors, volume 58 of Topics in Applied Physics. Springer, Berlin, 1985.
Gerard R. Richter, An optimal-order error estimate for the discontinuous Galerkin method, Math. Comp. 50 (1988), no. 181, 75–88. MR 917819, DOI 10.1090/S0025-5718-1988-0917819-3
Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439–471. MR 954915, DOI 10.1016/0021-9991(88)90177-5
Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31 (2006), no. 1-3, 417–429. MR 2209761, DOI 10.1080/03605300500361545
K. Tomizawa. Numerical simulation of sub micron semiconductor devices. Artech House, Boston, 1993.
Cédric Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 71–305. MR 1942465, DOI 10.1016/S1874-5792(02)80004-0
Cédric Villani, Hypocoercive diffusion operators, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 473–498. MR 2275692
Xiangxiong Zhang and Chi-Wang Shu, A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws, SIAM J. Numer. Anal. 48 (2010), no. 2, 772–795. MR 2670004, DOI 10.1137/090764384
Xiangxiong Zhang and Chi-Wang Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys. 229 (2010), no. 9, 3091–3120. MR 2601091, DOI 10.1016/j.jcp.2009.12.030
Xiangxiong Zhang and Chi-Wang Shu, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys. 229 (2010), no. 23, 8918–8934. MR 2725380, DOI 10.1016/j.jcp.2010.08.016
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
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.