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

Jin, Shi; Wang, Li

doi:10.1137/120886534

Cited by 13 publications

(8 citation statements)

References 34 publications

(57 reference statements)

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“…• AP schemes for other kinds of asymptotic behaviour. These include plasmas with a strong magnetic field and drift limits Lemou 2011, Hauck, Chacon anddel Castillo-Negrete 2014), highly oscillatory Vlasov-Poisson models (Crouseilles, Lemou and Mehats 2013) and high-field regimes for kinetic semiconductor equations (Jin and Wang 2013). We refer to the recent review by Degond (2013) for further examples.…”

Section: Discussionmentioning

confidence: 99%

Numerical methods for kinetic equations

Dimarco¹,

Pareschi²

2014

Acta Numerica

347

335

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In this survey we consider the development and the mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity.\ud \ud Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic preserving methods and the construction of hybrid schemes

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Section: Discussionmentioning

confidence: 99%

Numerical methods for kinetic equations

Dimarco¹,

Pareschi²

2014

Acta Numerica

347

335

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“…This form is convenient for designing AP schemes (Jin and Wang 2011) (Crouseilles and Lemou 2011), based on which one can easily use other well-developed AP frameworks. For more general collision operator, for example the semiconductor Boltzmann collision operator, this trick does not apply and one needs other ideas, for example the BGK penalization (Jin and Wang 2013). A variational approach was recently proposed in (Carrillo, Wang, Xu and Yan 2021), using the Wasserstein gradient structure, to get positivity and AP easily.…”

Section: High Electric Fieldmentioning

confidence: 99%

Asymptotic-Preserving Schemes for Multiscale Physical Problems

Jin¹

2021

Preprint

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We present the asymptotic transitions from microscopic to macroscopic physics, their computational challenges and the Asymptotic-Preserving (AP) strategies to efficiently compute multiscale physical problems. Specifically, we will first study the asymptotic transition from quantum to classical mechanics, from classical mechanics to kinetic theory, and then from kinetic theory to hydrodynamics. We then review some representative AP schemes that mimic, at the discrete level, these asymptotic transitions, hence can be used crossing scales and, in particular, capture the macroscopic behavior without resolving numerically the microscopic physical scale.

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“…A higher-order scheme was constructed in [8], which improved the strict parabolic stability condition to a hyperbolic one. An efficient AP scheme in the high field regime was developed in [25]. The authors in [16] further study the semiconductor Boltzmann equation with a two-scale stiff collision operators, by taking into account different effects including the interactions between electrons and the lattice defects caused by ionized impurities [3]; they design and demonstrate the efficiency and accuracy of an asymptoticpreserving scheme that leads to an energy-transport system as mean free path goes to zero at a discretized level.…”

Section: Introductionmentioning

confidence: 99%

A stochastic asymptotic-preserving scheme for the bipolar semiconductor Boltzmann-Poisson system with random inputs and diffusive scalings

Liu

2019

Journal of Computational Physics

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In this paper, we study the bipolar Boltzmann-Poisson model, both for the deterministic system and the system with uncertainties, with asymptotic behavior leading to the drift diffusion-Poisson system as the Knudsen number goes to zero. The random inputs can arise from collision kernels, doping profile and initial data. We adopt a generalized polynomial chaos approach based stochastic Galerkin (gPC-SG) method. Sensitivity analysis is conducted using hypocoercivity theory for both the analytical solution and the gPC solution for a simpler model that ignores the electric field, and it gives their convergence toward the global Maxwellian exponentially in time. A formal proof of the stochastic asymptotic-preserving (s-AP) property and a uniform spectral convergence with error exponentially decaying in time in the random space of the scheme is given. Numerical experiments are conducted to validate the accuracy, efficiency and asymptotic properties of the proposed method.

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

Cited by 13 publications

References 34 publications

Numerical methods for kinetic equations

Numerical methods for kinetic equations

Asymptotic-Preserving Schemes for Multiscale Physical Problems

A stochastic asymptotic-preserving scheme for the bipolar semiconductor Boltzmann-Poisson system with random inputs and diffusive scalings

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