136 research outputs found
A finite volume scheme for convection-diffusion equations with nonlinear diffusion derived from the Scharfetter-Gummel scheme
International audienceWe propose a finite volume scheme for convection-diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter-Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions
Numerical methods for drift-diffusion models
The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume discretization scheme and recent efforts to generalize this approach to general statistical distribution functions
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Numerical methods for drift-diffusion models
The van Roosbroeck system describes the semi-classical transport of
free electrons and holes in a self-consistent electric field using a
drift-diffusion approximation. It became the standard model to describe the
current flow in semiconductor devices at macroscopic scale. Typical devices
modeled by these equations range from diodes, transistors, LEDs, solar cells
and lasers to quantum nanostructures and organic semiconductors. The report
provides an introduction into numerical methods for the van Roosbroeck
system. The main focus lies on the Scharfetter-Gummel finite volume
disretization scheme and recent efforts to generalize this approach to
general statistical distribution functions
Convergence of a Finite Volume Scheme for a Corrosion Model
In this paper, we study the numerical approximation of a system of partial
dif-ferential equations describing the corrosion of an iron based alloy in a
nuclear waste repository. In particular, we are interested in the convergence
of a numerical scheme consisting in an implicit Euler scheme in time and a
Scharfetter-Gummel finite volume scheme in space
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On thermodynamic consistency of a Scharfetter-Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement
Driven by applications like organic semiconductors there is an
increased interest in numerical simulations based on drift-diffusion models
with arbitrary statistical distribution functions. This requires numerical
schemes that preserve qualitative properties of the solutions, such as
positivity of densities, dissipativity and consistency with thermodynamic
equilibrium. An extension of the Scharfetter-Gummel scheme guaranteeing
consistency with thermodynamic equilibrium is studied. It is derived by
replacing the thermal voltage with an averaged diffusion enhancement for
which we provide a new explicit formula. This approach avoids solving the
costly local nonlinear equations defining the current for generalized
Scharfetter-Gummel schemes
A finite volume scheme for nonlinear degenerate parabolic equations
We propose a second order finite volume scheme for nonlinear degenerate
parabolic equations. For some of these models (porous media equation,
drift-diffusion system for semiconductors, ...) it has been proved that the
transient solution converges to a steady-state when time goes to infinity. The
present scheme preserves steady-states and provides a satisfying long-time
behavior. Moreover, it remains valid and second-order accurate in space even in
the degenerate case. After describing the numerical scheme, we present several
numerical results which confirm the high-order accuracy in various regime
degenerate and non degenerate cases and underline the efficiency to preserve
the large-time asymptotic
Assessing the quality of the excess chemical potential flux scheme for degenerate semiconductor device simulation
The van Roosbroeck system models current flows in (non-)degenerate semiconductor devices. Focusing on the stationary model, we compare the excess chemical potential discretization scheme, a flux approximation which is based on a modification of the drift term in the current densities, with another state-of-the-art Scharfetter-Gummel scheme, namely the diffusion-enhanced scheme. Physically, the diffusion-enhanced scheme can be interpreted as a flux approximation which modifies the thermal voltage. As a reference solution we consider an implicitly defined integral flux, using Blakemore statistics. The integral flux refers to the exact solution of a local two point boundary value problem for the continuous current density and can be interpreted as a generalized Scharfetter-Gummel scheme. All numerical discretization schemes can be used within a Voronoi finite volume method to simulate charge transport in (non-)degenerate semiconductor devices. The investigation includes the analysis of Taylor expansions, a derivation of error estimates and a visualization of errors in local flux approximations to extend previous discussions. Additionally, drift-diffusion simulations of a p-i-n device are performed
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Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi-Dirac and Gauss-Fermi statistics
We compare three thermodynamically consistent ScharfetterGummel schemes
for different distribution functions for the carrier densities, including the
FermiDirac integral of order 1/2 and the GaussFermi integral. The most
accurate (but unfortunately also most costly) generalized ScharfetterGummel
scheme requires the solution of an integral equation. We propose a new method
to solve this integral equation numerically based on Gauss quadrature and
Newtons method. We discuss the quality of this approximation and plot the
resulting currents for FermiDirac and GaussFermi statistics. Finally, by
comparing two modified (diffusion-enhanced and inverse activity based)
ScharfetterGummel schemes with the more accurate generalized scheme, we show
that the diffusion-enhanced ansatz leads to considerably lower flux errors,
confirming previous results (J. Comp. Phys. 346:497-513, 2017)
A weighted Hybridizable Discontinuous Galerkin method for drift-diffusion problems
In this work we propose a weighted hybridizable discontinuous Galerkin method
(W-HDG) for drift-diffusion problems. By using specific exponential weights
when computing the product in each cell of the discretization, we are
able to replicate the behavior of the Slotboom change of variables, and
eliminate the drift term from the local matrix contributions. We show that the
proposed numerical scheme is well-posed, and numerically validates that it has
the same properties of classical HDG methods, including optimal convergence,
and superconvergence of postprocessed solutions. For polynomial degree zero,
dimension one, and vanishing HDG stabilization parameter, W-HDG coincides with
the Scharfetter-Gummel stabilized finite volume scheme (i.e., it produces the
same system matrix). The use of local exponential weights generalizes the
Scharfetter-Gummel stabilization (the state-of-the-art for Finite Volume
discretization of transport-dominated problems) to arbitrary high-order
approximations.Comment: 27 pages, 4 figures, 4 table
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