In this paper, we propose a finite-volume scheme for aggregation-diffusion equations based on a Scharfetter--Gummel approximation of the quadratic, nonlocal flux term. This scheme is analyzed concerning well-posedness and convergence towards solutions to the continuous problem. Also, it is proven that the numerical scheme has several structure-preserving features. More specifically, it is shown that the discrete solutions satisfy a free-energy dissipation relation analogous to the continuous model. Consequently, the numerical solutions converge in the large time limit to stationary solutions, for which we provide a thermodynamic characterization. Numerical experiments complement the study.
Electro-thermal transport phenomena in semiconductors are described by the non-isothermal drift-diffusion system. The equations take a remarkably simple form when assuming the Kelvin formula for the thermopower. We present a novel, non-isothermal generalization of the Scharfetter-Gummel finite volume discretization for degenerate semiconductors obeying Fermi-Dirac statistics, which preserves numerous structural properties of the continuous model on the discrete level. The approach is demonstrated by 2D simulations of a heterojunction bipolar transistor.
We present a Scharfetter-Gummel (SG) stabilization scheme for high-order Hybrid Discontinuous Galerkin (HDG) approximations of convection-diffusion problems. The scheme is based on a careful choice of the stabilization parameters used to define the numerical flux in the HDG method. We show that, in one dimension, the SG-HDG scheme is equivalent to the Finite Volume method stabilized with the Scharfetter--Gummel on the dual grid, for all orders of HDG schemes.
Many challenges faced in today's semiconductor devices are related to self-heating phenomena. The optimization of device designs can be assisted by numerical simulations using the non-isothermal drift-diffusion system, where the magnitude of the thermoelectric cross effects is controlled by the Seebeck coefficient. We show that the model equations take a remarkably simple form when assuming the so-called Kelvin formula for the Seebeck coefficient. The corresponding heat generation rate involves exactly the three classically known self-heating effects, namely Joule, recombination and Thomson-Pe…
We introduce a family of various finite volume discretization schemes for the Fokker--Planck operator, which are characterized by different weight functions on the edges. This family particularly includes the well-established Scharfetter--Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second pla…
In this work, we consider an advection-diffusion equation, coupled to a Poisson equation for the velocity field. This type of coupling is typically encountered in models arising from plasma physics or porous media flow. The aim of this work is to build upon the complete flux scheme (an improvement over the Scharfetter-Gummel scheme by considering the contribution of the source term), so that its second-order convergence, which is uniform in Péclet numbers, carries over to these models. This is done by considering a piecewise linear approximation of the velocity field, which is then used for de…
Exponential decay of a finite volume scheme to the thermal equilibrium for drift--diffusion systems
Univ. Nantes, CNRS, UMR 6629 - Laboratoire Jean Leray（南特大学，法国国家科学研究中心，联合研究单位6629-让·勒雷实验室）；Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，联合研究单位8524-保罗·潘勒韦实验室）
Study of a fully implicit scheme for the drift-diffusion system. Asymptotic behavior in the quasi-neutral limit
Université de Nantes（南特大学）；Laboratoire de Mathématiques Jean Leray, UMR 6629 - CNRS（让·勒雷数学实验室，UMR 6629 - 法国国家科学研究中心）；Université Lille 1（里尔第一大学）；Laboratoire P. Painlevé, CNRS UMR 8524（P. 庞加莱数学实验室，CNRS UMR 8524）；INRIA Lille Nord Europe（欧洲北部法国国家信息与自动化研究所）；Institut de Mathématiques de Toulouse, CNRS UMR 5219（图卢兹数学研究所，CNRS UMR 5219）；Université de Toulouse（图卢兹大学）
A new model for simulating heat, air and moisture transport in porous building materials
LOCIE–CNRS, Université Savoie Mont Blanc（萨瓦-蒙布朗大学）；LAMA–CNRS, Université Savoie Mont Blanc（萨瓦-蒙布朗大学）；Pontifical Catholic University of Paraná（巴拉那天主教大学）；IIT University（阿拉木图工业大学）
