A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson equations

Cheng, Yingda; Gamba, Irene M.; Majorana, Armando; Shu, Chi-Wang

doi:10.1007/bf03322587

Cited by 22 publications

(13 citation statements)

References 23 publications

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“…The spatial gradient ∇ x f s (t,x,k) can be discretized in the same way as the drift one. In this case the discretization depends on the geometry of the spatial domain and boundary conditions (see for instance [13,14]). Since in this paper we consider only space homogeneous solutions, the details are skipped.…”

Section: The Numerical Methodsmentioning

confidence: 99%

“…Direct solutions of the electron transport equations with finite difference methods have been obtained in [3] while a Discontinuous Galerkin (DG) method has been used in [8,11,12]. See [13,14] for application of the DG method to traditional semiconductors, while numerical schemes for the Wigner equation can be found in [15]. A hydrodynamical model based on the maximum entropy principle (MEP) has been formulated in [16] using a set of field variables which proved to be successful for traditional semiconductors as silicon [17][18][19][20][21][22][23], gallium arsenide [17,24], silicon carbide [25].…”

Section: Introductionmentioning

confidence: 99%

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Simulation of Bipolar Charge Transport in Graphene by Using a Discontinuous Galerkin Method

Majorana¹

2019

CiCP

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Charge transport in suspended monolayer graphene is simulated by a numerical deterministic approach, based on a discontinuous Galerkin (DG) method, for solving the semiclassical Boltzmann equation for electrons. Both the conduction and valence bands are included and the interband scatterings are taken into account. The use of a Direct Simulation Monte Carlo (DSMC) approach, which properly describes the interband scatterings, is computationally very expensive because the valence band is very populated and a huge number of particles is needed. Also the choice of simulating holes instead of electrons does not overcome the problem because there is a certain degree of ambiguity in the generation and recombination terms of electronhole pairs. Often, direct solutions of the Boltzmann equations with a DSMC neglect the interband scatterings on the basis of physical arguments. The DG approach does not suffer from the previous drawbacks and requires a reasonable computing effort. In the present paper the importance of the interband scatterings is accurately evaluated for several values of the Fermi energy, addressing the issue related to the validity of neglecting the generation-recombination terms. It is found out that the inclusion of the interband scatterings produces huge variations in the average values, as the current, with zero Fermi energy while, as expected, the effect of the interband scattering becomes negligible by increasing the absolute value of the Fermi energy.

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Section: The Numerical Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simulation of Bipolar Charge Transport in Graphene by Using a Discontinuous Galerkin Method

Majorana¹

2019

CiCP

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“…The high order accuracy and locality of data in particular make our approach especially advantageous; higher order polynomials provide a level of accuracy equivalent to refining the grid at a fraction of the cost, and the locality of data significantly reduces the amount of communication required in the update, enabling the algorithm to scale to well on both standard computing architecture and many-core devices. In fact, the discontinuous Galerkin algorithm has been gaining increased attention as a means of discretizing high dimensional transport equations and various flavors of the Vlasov equation, including Vlasov-Poisson, Vlasov-Ampere, and the aforementioned Vlasov-Maxwell [28,29,30].…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin algorithms for fully kinetic plasmas

Juno¹,

Hakim²,

TenBarge³

et al. 2018

Journal of Computational Physics

111

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We present a new algorithm for the discretization of the Vlasov-Maxwell system of equations for the study of plasmas in the kinetic regime. Using the discontinuous Galerkin finite element method for the spatial discretization, we obtain a high order accurate solution for the plasma's distribution function. Time stepping for the distribution function is done explicitly with a third order strong-stability preserving Runge-Kutta method. Since the Vlasov equation in the Vlasov-Maxwell system is a high dimensional transport equation, up to six dimensions plus time, we take special care to note various features we have implemented to reduce the cost while maintaining the integrity of the solution, including the use of a reduced high-order basis set. A series of benchmarks, from simple wave and shock calculations, to a five dimensional turbulence simulation, are presented to verify the efficacy of our set of numerical methods, as well as demonstrate the power of the implemented features

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“…In the case of Boltzmann equation, there is a very rich literature. Both probabilistic approaches such as direct simulation Monte Carlo [39,40], as well as deterministic methods, e.g., discontinuous Galerkin and spectral methods [41,42,43], have been proposed to compute the solution. Probabilistic methods such as direct Monte Carlo are extensively used because of their very low computational cost compared to finite-volumes, finite-differences or spectral methods, especially in the multi-dimensional case.…”

Section: Long-term Integrationmentioning

confidence: 99%

Numerical methods for high-dimensional probability density function equations

Cho

Venturi

Karniadakis

2016

Journal of Computational Physics

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In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker-Planck and Dostupov-Pugachev equations), random wave theory (Malakhov-Saichev equations) and coarse-grained stochastic systems (Mori-Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables.

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A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson equations

Cited by 22 publications

References 23 publications

Simulation of Bipolar Charge Transport in Graphene by Using a Discontinuous Galerkin Method

Simulation of Bipolar Charge Transport in Graphene by Using a Discontinuous Galerkin Method

Discontinuous Galerkin algorithms for fully kinetic plasmas

Numerical methods for high-dimensional probability density function equations

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