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Entropy-stable positivity-preserving DG schemes for Boltzmann-Poisson models of collisional electronic transport along energy bands

Jose A. Morales Escalante, Irene M. Gamba

TL;DR

The paper develops entropy-stable, positivity-preserving discontinuous Galerkin schemes for Boltzmann-Poisson models of collisional electronic transport along semiconductor energy bands in curvilinear momentum coordinates. It leverages a Hamiltonian based entropy weight $e^H$ with $H=\varepsilon(\vec{p})-q\Phi$ and uses a Jacobian $J$ to formulate a divergence structure that yields discrete entropy dissipation and stability for both 1D and higher dimensional problems, including periodic and specular boundaries. Positivity is enforced by treating collisions as a source and applying convex decompositions and Zhang Shu limiters to preserve positive cell averages and, when needed, pointwise positivity. The work also provides error estimates for semi-discrete DG schemes and discusses the extension to 2Dx-3DK configurations with specular boundaries, making the approach applicable to hot electron transport in nanoscale devices with energy band structure effects.

Abstract

This work is related to developing entropy-stable positivity-preserving Discontinuous Galerkin (DG) methods as a computational scheme for Boltzmann-Poisson systems modeling the probability density of collisional electronic transport along semiconductor energy bands. In momentum coordinates representing spherical / energy-angular variables, we pose the respective Vlasov-Boltzmann equation with a linear collision operator and a singular measure, modeling scatterings as functions of the band structure appropriately for hot electron nanoscale transport. We show stability results of semi-discrete DG schemes under an entropy norm for 1D-position (2D-momentum) and 2D-position (3D-momentum), using dissipative properties of the collisional operator given its entropy inequality. The latter depends on an exponential of the Hamiltonian rather than the Maxwellian associated with only kinetic energy. For the 1D problem, knowing the analytic solution to the Poisson equation and convergence to a constant current is crucial to obtaining full stability (weighted entropy norm decreasing over time). For the 2D problem, specular reflection boundary conditions and periodicity are considered in estimating stability under an entropy norm. Regarding the positivity-preservation proofs in the DG scheme for the 1D problem, inspired by \cite{ZhangShu1}, \cite{ZhangShu2}, and \cite{CGP}, \cite{EECHXM-JCP}, we treat collisions as a source and find convex combinations of the transport and collision terms which guarantee positivity of the cell average of our numerical probability density at the next time. The positivity of the numerical solution to the probability density in the domain is guaranteed by applying the limiters in \cite{ZhangShu1} and \cite{ZhangShu2} that preserve the cell average modifying the slope of the piecewise linear solutions to make the function non-negative.

Entropy-stable positivity-preserving DG schemes for Boltzmann-Poisson models of collisional electronic transport along energy bands

TL;DR

The paper develops entropy-stable, positivity-preserving discontinuous Galerkin schemes for Boltzmann-Poisson models of collisional electronic transport along semiconductor energy bands in curvilinear momentum coordinates. It leverages a Hamiltonian based entropy weight with and uses a Jacobian to formulate a divergence structure that yields discrete entropy dissipation and stability for both 1D and higher dimensional problems, including periodic and specular boundaries. Positivity is enforced by treating collisions as a source and applying convex decompositions and Zhang Shu limiters to preserve positive cell averages and, when needed, pointwise positivity. The work also provides error estimates for semi-discrete DG schemes and discusses the extension to 2Dx-3DK configurations with specular boundaries, making the approach applicable to hot electron transport in nanoscale devices with energy band structure effects.

Abstract

This work is related to developing entropy-stable positivity-preserving Discontinuous Galerkin (DG) methods as a computational scheme for Boltzmann-Poisson systems modeling the probability density of collisional electronic transport along semiconductor energy bands. In momentum coordinates representing spherical / energy-angular variables, we pose the respective Vlasov-Boltzmann equation with a linear collision operator and a singular measure, modeling scatterings as functions of the band structure appropriately for hot electron nanoscale transport. We show stability results of semi-discrete DG schemes under an entropy norm for 1D-position (2D-momentum) and 2D-position (3D-momentum), using dissipative properties of the collisional operator given its entropy inequality. The latter depends on an exponential of the Hamiltonian rather than the Maxwellian associated with only kinetic energy. For the 1D problem, knowing the analytic solution to the Poisson equation and convergence to a constant current is crucial to obtaining full stability (weighted entropy norm decreasing over time). For the 2D problem, specular reflection boundary conditions and periodicity are considered in estimating stability under an entropy norm. Regarding the positivity-preservation proofs in the DG scheme for the 1D problem, inspired by \cite{ZhangShu1}, \cite{ZhangShu2}, and \cite{CGP}, \cite{EECHXM-JCP}, we treat collisions as a source and find convex combinations of the transport and collision terms which guarantee positivity of the cell average of our numerical probability density at the next time. The positivity of the numerical solution to the probability density in the domain is guaranteed by applying the limiters in \cite{ZhangShu1} and \cite{ZhangShu2} that preserve the cell average modifying the slope of the piecewise linear solutions to make the function non-negative.

Paper Structure

This paper contains 17 sections, 8 theorems, 318 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: On the problem of energy - curvilinear coordinates
  3. Stability of DG schemes under entropy norms
  4. 1Dx-2Dp Diode Symmetric Problem
  5. RKDG Entropy preserving scheme for the 1Dx-2Dp Boltzmann-Poisson system
  6. The Cut-off-domain
  7. The DG-FEM scheme formulation for the transformed Boltzmann equation in the $(x,p,\mu)$-space computational domain.
  8. Stability of DG scheme under entropy norm for Periodic Boundary Conditions
  9. 2DX-3DK Problem: DG scheme and Stability under an entropy norm for Periodic and Reflective Boundary Conditions
  10. Specular Reflection Boundaries
  11. Internal Edges and Periodic Boundaries
  12. Error estimates for semi-discrete DG scheme with curvilinear momentum coordinates
  13. Conclusions
  14. Acknowledgements
  15. Positivity Preservation in DG Scheme for BP
  16. ...and 2 more sections

Key Result

Theorem 1

(Stability under the entropy norm $\int f_h \, g_h e^H \, p^2 dp d\mu dx$ for a given periodic potential $\Phi(x,t)$): Consider the semi-discrete solution $f_h$ to the DG formulation in (entropyPPDGform) for the BP system in momentum curvilinear coordinates. We have then that

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Theorem 6
  • ...and 4 more