A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson equations
We are interested in the deterministic computation of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The main difficulty of such computation arises from the very high dimensions of the model, making it necessary to use relatively coarse meshes and hence requiring the numerical solver to be stable and to have good resolution under coarse meshes. In this paper we give a brief survey of the discontinuous Galerkin (DG) method, which is a finite element method using discontinuous piecewise polynomials as basis functions and numerical fluxes based on upwinding for stability, for solving the Boltzmann-Poisson system. In many situations, the deterministic DG solver can produce accurate solutions with equal or less CPU time than the traditional DSMC (Direct Simulation Monte Carlo) solvers. In order to make the presentation more concise and to highlight the main ideas of the algorithm, we use a simplified model to describe the details of the DG method. Sample simulation results on the full Boltzmann-Poisson system are also given.
KeywordsCoarse Mesh Discontinuous Galerkin Discontinuous Galerkin Method Direct Simulation Monte Carlo Boltzmann Transport Equation
Unable to display preview. Download preview PDF.
- 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, in Transport Phenomena and Kinetic Theory Applications to Gases, Semiconductors, Photons, and Biological Systems. C. Cercignani and E. Gabetta (Eds.), Birkhäuser (2006), pp. 151–171.Google Scholar
- Y. Cheng, I.M. Gamba, A. Majorana and C.-W. Shu, Discontinuous Galerkin solver for the semiconductor Boltzmann equation, SISPAD 07, T. Grasser and S. Selberherr, editors, Springer (2007), pp. 257–260.Google Scholar
- Y. Cheng, I. Gamba, A. Majorana and C.-W. Shu, A discontinuous Galerkin solver for full-band Boltzmann-Poisson models, the Proceeding of IWCE 13, pp. 211–214, 2009.Google Scholar
- Y. Cheng, I.M. Gamba, A. Majorana and C.-W. Shu, High order positive discontinuous Galerkin schemes for the Boltzmann-Poisson system with full bands, in preparation.Google Scholar
- Y. Cheng, I.M. Gamba and J. Proft, Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations, Mathematics of Computation, to appear.Google Scholar
- M. L. Cohen and J. Chelikowsky. Electronic Structure and Optical Properties of Semiconductors. Springer-Verlag, 1989.Google Scholar
- K. Tomizawa, Numerical Simulation of Submicron Semiconductor Devices, Artech House: Boston, 1993.Google Scholar
- M. C. Vecchi, D. Ventura, A. Gnudi and G. Baccarani. Incorporating full band-structure effects in spherical harmonics expansion of the Boltzmann transport equation, in Proceedings of NUPAD V Conference, 8 (1994), pp. 55–58.Google Scholar
- J.M. Ziman, Electrons and Phonons. The Theory of Transport Phenomena in Solids, Oxford University Press: Oxford, 2000.Google Scholar