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

TL;DR

This paper develops a stochastic asymptotic-preserving scheme for the bipolar Boltzmann-Poisson system with uncertainties, ensuring accurate and efficient simulations across different regimes and random inputs, validated by numerical experiments.

Contribution

It introduces a novel stochastic AP scheme using gPC-SG for the bipolar Boltzmann-Poisson system with uncertainties, proving its asymptotic-preserving and spectral convergence properties.

Findings

The scheme accurately captures the asymptotic limit as Knudsen number approaches zero.

Numerical results confirm the scheme's efficiency and exponential convergence in random space.

The method effectively handles uncertainties in collision kernels, doping, and initial data.

Abstract

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.