Discontinuous Galerkin methods for the Boltzmann-Poisson systems in semiconductor device simulations

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

doi:10.1063/1.3562758

Cited by 6 publications

(9 citation statements)

References 16 publications

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“…The singular nature of s(v, v ) makes the collision hard to compute numerically, but the cylindrical symmetry of s makes it possible to use polar coordinates so that the singularity in the delta function can be removed and the dimension of the integral can be decreased by one [5,6,8]. However, this trick is not easy to implement here since we treat the stiff force term implicitly, and changing to polar coordinates will make it harder to invert.…”

Section: The Degenerate Casementioning

confidence: 95%

“…This method can be easily extended to the case with a nonparabolic energy diagram such as Kane's model [5,6,8] since the convection term is treated explicitly.…”

Section: The Nondegenerate Anisotropic Casementioning

confidence: 98%

“…In the one-dimensional examples, we use the following settings unless otherwise specified. The computational domain for x and v is [0, 8,8] with N x = 128 and N v = 32. The time step is chosen Downloaded 12/01/14 to 141.209.144.122.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…It was revisited in [2], where the convergence to entropy solutions and existence of shock profiles for the limit nonlinear conservation law were considered. Considerable literature has been devoted to the design of efficient and accurate numerical methods for (1.1), such as [20,5,6,8,13], to name just a few. These schemes become inefficient in the high field regime.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic-Preserving Numerical Schemes for the Semiconductor Boltzmann Equation Efficient in the High Field Regime

Jin¹,

Wang²

2013

SIAM J. Sci. Comput.

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We present asymptotic-preserving numerical schemes for the semiconductor Boltzmann equation efficient in the high field regime. A major challenge in this regime is that there may be no explicit expression of the local equilibrium which is the main component of classical asymptoticpreserving schemes. ], our idea is to penalize the stiff collision term with a classical Bhatnagar-Gross-Krook operator-which is not the local equilibrium in the high field limit-while treating the stiff force term implicitly with the spectral method. These schemes, despite being implicit, can be inverted easily with a stability independent of the physically small parameter. We design these schemes for both nondegenerate and degenerate cases and show their asymptotic properties. We present several numerical examples to validate the efficiency, accuracy, and asymptotic properties of these schemes.

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Section: The Degenerate Casementioning

confidence: 95%

“…This method can be easily extended to the case with a nonparabolic energy diagram such as Kane's model [5,6,8] since the convection term is treated explicitly.…”

Section: The Nondegenerate Anisotropic Casementioning

confidence: 98%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic-Preserving Numerical Schemes for the Semiconductor Boltzmann Equation Efficient in the High Field Regime

Jin¹,

Wang²

2013

SIAM J. Sci. Comput.

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“…For problems of charge transport in semiconductor devices, DG methods are very promising and have provided accurate results at a comparable computational cost [19,20]. It seems, DG could be a potential method for kinetic equations.…”

Section: Introductionmentioning

confidence: 99%

A conservative discontinuous Galerkin scheme with O(N2) operations in computing Boltzmann collision weight matrix

Gamba¹,

Zhang²

2014

AIP Conference Proceedings

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A Runge-Kutta discontinuous Galerkin solver for 2D Boltzmann model equations: Verification and analysis of computational performance AIP Conf. Proc. 1501, 381 (2012) Abstract. In the present work, we propose a deterministic numerical solver for the homogeneous Boltzmann equation based on Discontinuous Galerkin (DG) methods. The weak form of the collision operator is approximated by a quadratic form in linear algebra setting. We employ the property of "shifting symmetry" in the weight matrix to reduce the computing complexity from theoretical O(N 3 ) down to O(N 2 ), with N the total number of freedom for d-dimensional velocity space. In addition, the sparsity is also explored to further reduce the storage complexity. To apply lower order polynomials and resolve loss of conserved quantities, we invoke the conservation routine at every time step to enforce the conservation of desired moments (mass, momentum and/or energy), with only linear complexity. Due to the locality of the DG schemes, the whole computing process is well parallelized using hybrid OpenMP and MPI. The current work only considers integrable angular cross-sections under elastic and/or inelastic interaction laws. Numerical results on 2-D and 3-D problems are shown.

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A discontinuous Galerkin method for the Vlasov–Poisson system

Heath

Gamba

Morrison

et al. 2012

Journal of Computational Physics

114

119

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A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates associated system's approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov-Poisson system.

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Discontinuous Galerkin methods for the Boltzmann-Poisson systems in semiconductor device simulations

Cited by 6 publications

References 16 publications

Asymptotic-Preserving Numerical Schemes for the Semiconductor Boltzmann Equation Efficient in the High Field Regime

Asymptotic-Preserving Numerical Schemes for the Semiconductor Boltzmann Equation Efficient in the High Field Regime

A conservative discontinuous Galerkin scheme with O(N2) operations in computing Boltzmann collision weight matrix

A discontinuous Galerkin method for the Vlasov–Poisson system

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