We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

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Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

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The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobilike equations.

Review Article Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.

Runge-Kutta discontinuous Galerkin methods for convection-dominated problems

The classical hydrodynamic equations can be extended to include quantum effects by incorporating the first quantum corrections These quantum corrections are $O( {\hbar ^2 } )$ The full three-dimensional quantum hydrodynamic (QHD) model is derived for the first time by a moment expansion of the Wigner–Boltzmann equation The QHD conservation laws have the same form as the classical hydrodynamic equations, but the energy density and stress tensor have additional quantum terms These quantum terms allow particles to tunnel through potential barriers and to build up in potential wellsThe three-dimensional QHD transport equations are mathematically classified as having two Schrodinger modes, two hyperbolic modes, and one parabolic mode The one-dimensional steady-state QHD equations are discretized in conservation form using the second upwind methodSimulations of a resonant tunneling diode are presented that show charge buildup in the quantum well and negative differential resistance (NDR) in the current-v

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The quantum hydrodynamic model for semiconductor devices

An effective stress tensor and energy density for the quantum hydrodynamic (QHD) equations are derived in the Born approximation to the Bloch equation. The quantum potential appearing in the stress tensor and energy density is valid to all orders of ${\mathrm{\ensuremath{\Elzxh}}}^{2}$ and to first order in \ensuremath{\beta}V, and involves both a smoothing integration of the classical potential over space and an averaging integration over temperature. In the presence of discontinuities in the classical potential (which occur, for example, at potential barriers in semiconductors), the effective stress tensor and energy density are more tractable analytically and numerically than in the original O(${\mathrm{\ensuremath{\Elzxh}}}^{2}$) QHD theory. By cancelling the leading singularity in the classical potential at a barrier and leaving a residual smooth effective potential (with a lower potential height) in the barrier region, the effective stress tensor makes the barrier partially transparent to the particle flow and provides the mechanism for particle tunneling in the QHD model. \textcopyright{} 1996 The American Physical Society.

Smooth quantum potential for the hydrodynamic model

An introduction to the hydrodynamic model for semiconductor devices is presented. Special attention is paid to classifying the hydrodynamic PDEs (partial differential equations) and analyzing their nonlinear wave structure. Numerical simulations of the ballistic diode using the hydrodynamical device model are presented, as an illustrative elliptic problem. The importance of nonlinear block iterative methods is emphasized. Arguments for existence of solutions and convergence of numerical methods are given for the case of subsonic electron flow. >

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Numerical methods for the hydrodynamic device model: subsonic flow

Appropriate numerical methods for steady-state simulations (including shock waves) when the electron flow is both subsonic and supersonic are addressed. The one-dimensional steady-state hydrodynamic equations will then be elliptic in the subsonic regions and hyperbolic/elliptic in the supersonic regions. A second upwind method is used for both elliptic and hyperbolic/elliptic regions. In the elliptic regions, the second upwind method is related to the Scharfetter-Gummell exponential fitting method. The hydrodynamic model consists of a set of nonlinear conservation laws for particle number, momentum, and energy, coupled to Poisson's equation for the electric potential. The nonlinear conservation laws are just the Euler equations of gas dynamics for a gas of charged particles in an electric field, with the addition of a heat conduction term. Thus the hydrodynamic model partial differential equations (PDEs) have hyperbolic, parabolic, and elliptic modes. The nonlinear hyperbolic modes support shock waves. The first numerical simulations of a steady-state electron shock wave in a semiconductor device are presented, using the hydrodynamic model. For the ballistic diode (which models the channel of a MOSFET), the shock wave is fully developed in Si (with 1-V bias) at 300 K for a 0.1- mu m channel and at 77 K for a 1.0- mu m channel. >

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Numerical simulation of a steady-state electron shock wave in a submicrometer semiconductor device

A statistical model is analyzed for the growth of bubbles in a Rayleigh–Taylor unstable interface. The model is compared to solutions of the full Euler equations for compressible two phase flow, using numerical solutions based on the method of front tracking. The front tracking method has the distinguishing feature of being a predominantly Eulerian method in which sharp interfaces are preserved with zero numerical diffusion. Various regimes in the statistical model exhibiting qualitatively distinct behavior are explored.

Carl L. Gardner

Papers

The quantum hydrodynamic model for semiconductor devices

Smooth quantum potential for the hydrodynamic model

Numerical methods for the hydrodynamic device model: subsonic flow

Numerical simulation of a steady-state electron shock wave in a submicrometer semiconductor device

The dynamics of bubble growth for Rayleigh-Taylor unstable interfaces