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Asymptotic-preserving and positivity-preserving discontinuous Galerkin method for the semiconductor Boltzmann equation in the diffusive scaling

Huan Ding, Liu Liu, Xinghui Zhong

TL;DR

This work develops an asymptotic-preserving and positivity-preserving discontinuous Galerkin method for the linear semiconductor Boltzmann equation under diffusive scaling. By employing an even-odd decomposition to obtain a diffusive-relaxation system and coupling a robust implicit relaxation step with a third-order SSP Runge-Kutta transport step to a high-order DG spatial discretization, the scheme remains accurate across kinetic and diffusive regimes. A positivity-preserving limiter ensures physical distribution values while a rigorous stability analysis within the even-odd framework confirms CFL-bounded energy decay, and an AP analysis shows convergence to a drift-diffusion limit as $\varepsilon\to0$. Numerical experiments validate high-order spatial accuracy, AP property, and effective handling of mixed regimes, highlighting the method’s potential for efficient semiconductor device simulations in multiscale settings.

Abstract

In this paper, we develop an asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) method for solving the semiconductor Boltzmann equation in the diffusive scaling. We first formulate the diffusive relaxation system based on the even-odd decomposition method, which allows us to split into one relaxation step and one transport step. We adopt a robust implicit scheme that can be explicitly implemented for the relaxation step that involves the stiffness of the collision term, while the third-order strong-stability-preserving Runge-Kutta method is employed for the transport step. We couple this temporal scheme with the DG method for spatial discretization, which provides additional advantages including high-order accuracy, $h$-$p$ adaptivity, and the ability to handle arbitrary unstructured meshes. A positivity-preserving limiter is further applied to preserve physical properties of numerical solutions. The stability analysis using the even-odd decomposition is conducted for the first time. We demonstrate the accuracy and performance of our proposed scheme through several numerical examples.

Asymptotic-preserving and positivity-preserving discontinuous Galerkin method for the semiconductor Boltzmann equation in the diffusive scaling

TL;DR

This work develops an asymptotic-preserving and positivity-preserving discontinuous Galerkin method for the linear semiconductor Boltzmann equation under diffusive scaling. By employing an even-odd decomposition to obtain a diffusive-relaxation system and coupling a robust implicit relaxation step with a third-order SSP Runge-Kutta transport step to a high-order DG spatial discretization, the scheme remains accurate across kinetic and diffusive regimes. A positivity-preserving limiter ensures physical distribution values while a rigorous stability analysis within the even-odd framework confirms CFL-bounded energy decay, and an AP analysis shows convergence to a drift-diffusion limit as . Numerical experiments validate high-order spatial accuracy, AP property, and effective handling of mixed regimes, highlighting the method’s potential for efficient semiconductor device simulations in multiscale settings.

Abstract

In this paper, we develop an asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) method for solving the semiconductor Boltzmann equation in the diffusive scaling. We first formulate the diffusive relaxation system based on the even-odd decomposition method, which allows us to split into one relaxation step and one transport step. We adopt a robust implicit scheme that can be explicitly implemented for the relaxation step that involves the stiffness of the collision term, while the third-order strong-stability-preserving Runge-Kutta method is employed for the transport step. We couple this temporal scheme with the DG method for spatial discretization, which provides additional advantages including high-order accuracy, - adaptivity, and the ability to handle arbitrary unstructured meshes. A positivity-preserving limiter is further applied to preserve physical properties of numerical solutions. The stability analysis using the even-odd decomposition is conducted for the first time. We demonstrate the accuracy and performance of our proposed scheme through several numerical examples.

Paper Structure

This paper contains 16 sections, 6 theorems, 82 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model equations and temporal discretization
  3. Even-odd decomposition and diffusive relaxation system
  4. Temporal discretization
  5. Fully discrete numerical scheme
  6. Discontinuous Galerkin method
  7. Positivity-preserving limiter
  8. Boundary conditions
  9. Theoretical analysis
  10. Stability analysis
  11. Preliminaries
  12. Main result
  13. Analysis of asymptotic-preserving property
  14. Numerical examples
  15. Conclusion
  16. ...and 1 more sections

Key Result

Lemma 1

\newlabellemma:inverse0 For any functions $u\in\mathbb{V}_h^k$, there holds for $i=1,\dots, N_x$, where $C_t, C_i$ are positive constants independent of $u$, $I_i$, and $h$.

Figures (6)

  • Figure 1: Numerical solutions $\rho$ for the kinetic and diffusive regimes of Example \ref{['eg:fixed_E']}.
  • Figure 2: Convergence to the drift-diffusion system for Example \ref{['eg:fixed_E']}.
  • Figure 3: Time evolution of numerical solutions $f_h$ for the kinetic (left) and diffusive (right) regimes of Example \ref{['eg:fixed_E']}, with $T=0.05$ (top), $T=0.5$ (middle), and $T=5$ (bottom).
  • Figure 4: Numerical results for the Boltzmann-Poisson system in Example \ref{['eg:BPSys']}.
  • Figure 5: Electric field $E$ (left) and electric potential $\Phi$ (right) for Boltzmann-Poisson system and the limiting drift-diffusion system in Example \ref{['eg:BPSys']}.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Lemma 1: inverse inequality
  • Lemma 2
  • Theorem 3: Stability
  • Remark 4.1
  • Lemma 4
  • Proof 1
  • Lemma 5
  • Proof 2
  • Lemma 6
  • Proof 3: Proof of Main Theorem \ref{['Thm:main']}
  • ...and 4 more