There can be little doubt that the Monte Carlo method for semiconductor device simulation has enormous power as a research tool. It represents a detailed physical model of the semiconductor material(s), and provides a high degree of insight into the microscopic transport processes. However, if the authority ascribed to Monte Carlo models of devices at 1/spl mu/m feature size is to be maintained for devices below O.1/spl mu/m, modelling of the fundamental physics must be further improved. And if the Monte Carlo method is to be successful as a semiconductor device design tool, the device model must be made more realistic. Success in the industrial sector depends on this, but also on achieving fast run-times optimisation - where the scope and need for ingenuity is now greatest.

The Monte Carlo method for semiconductor device simulation

Hamilton-Jacobi equations are frequently encountered in applications,e.g.,in control theory and differential games.Hamilton-Jacobi equations are closely related to hyperbolic conservation laws-in one space dimension the former is simply the integrated version of the latter.Similarity also exists for the multidimensional cases,and this is helpful in designing difference approximations.In this paper central weighted essentially non-oscillatory (CWENO) schemes for Hamilton-Jacobi equations are investigated,which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives.The schemes are numerically tested on a variety of one-dimensional problems,including a problem related to control optimization.High-order accuracy in smooth regions,high resolution of discontinuities in the derivatives,and convergence to viscosity solutions are observed.

High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

https://hal.archives-ouvertes.fr/hal-00986714/document

Numerical methods for kinetic equations

In an earlier study (Zhang & Shu 2010 b J. Comput. Phys. 229 , 3091–3120 ([doi:10.1016/j.jcp.2009.12.030][1])), genuinely high-order accurate finite volume and discontinuous Galerkin schemes satisfying a strict maximum principle for scalar conservation laws were developed. The main advantages of such schemes are their provable high-order accuracy and their easiness for generalization to multi-dimensions for arbitrarily high-order schemes on structured and unstructured meshes. The same idea can be used to construct high-order schemes preserving the positivity of certain physical quantities, such as density and pressure for compressible Euler equations, water height for shallow water equations and density for Vlasov–Boltzmann transport equations. These schemes have been applied in computational fluid dynamics, computational astronomy and astrophysics, plasma simulation, population models and traffic flow models. In this paper, we first review the main ideas of these maximum-principle-satisfying and positivity-preserving high-order schemes, then present a simpler implementation which will result in a significant reduction of computational cost especially for weighted essentially non-oscillatory finite-volume schemes.

 [1]: http://dx.doi.org/10.1016/j.jcp.2009.12.030

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2011.0153

Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments

Criteria useful for the investigation of the stability of a system of charged particles are derived from the Boltzmann equation in the small m/e limit. These criteria are obtained from the examination of the variation of the energy due to a perturbation, when subject to the general constraint that all regular, time‐independent constants of the motion (including the energy) have their equilibrium values.The first‐order variation of the energy vanishes trivially, and the second‐order variation yields a quadratic form in the displacement variable ξ (which may be introduced because of the well‐known properties of this limit). The positive‐definiteness of this form is a sufficient condition for stability.Several theorems are stated comparing stability under the present theory with that under conventional hydromagnetic fluid theories where heat flow along magnetic lines of force is neglected. Generalizations can be made to systems where the effect of collisions is included.

On the Stability of Plasma in Static Equilibrium

In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal (k + 1)-th order of accuracy when using piecewise k-th degree polynomials, under the condition that k + 1 is greater than or equal to the order of the equation.

/pdf/a-discontinuous-galerkin-finite-element-method-for-time-3xpcplzvun.pdf

A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

In this paper, we study the superconvergence property for the discontinuous Galerkin (DG) and the local discontinuous Galerkin (LDG) methods for solving one-dimensional time dependent linear conservation laws and convection-diffusion equations. We prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection-diffusion equations. The order of superconvergence for both cases is proved to be $k+\frac{3}{2}$ when piecewise $P^k$ polynomials with $k\geq1$ are used. The proof is valid for arbitrary nonuniform regular meshes and for piecewise $P^k$ polynomials with arbitrary $k\geq1$, improving upon the results in [Y. Cheng and C.-W. Shu, J. Comput. Phys., 227 (2008), pp. 9612-9627], [Y. Cheng and C.-W. Shu, Computers and Structures, 87 (2009), pp. 630-641] in which the proof based on Fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise $P^1$ polynomials.

/pdf/superconvergence-of-discontinuous-galerkin-and-local-1d5gjygnu5.pdf

Superconvergence of Discontinuous Galerkin and Local Discontinuous Galerkin Schemes for Linear Hyperbolic and Convection-Diffusion Equations in One Space Dimension

https://users.math.msu.edu/users/ycheng/papers/hj_jcp.pdf

A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations

https://users.math.msu.edu/users/ycheng/papers/CMAME.pdf

A discontinuous Galerkin solver for Boltzmann–Poisson systems in nano devices

https://users.math.msu.edu/users/ycheng/papers/superconvergence_jcp.pdf

Yingda Cheng

Papers

A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

Superconvergence of Discontinuous Galerkin and Local Discontinuous Galerkin Schemes for Linear Hyperbolic and Convection-Diffusion Equations in One Space Dimension

A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations

A discontinuous Galerkin solver for Boltzmann–Poisson systems in nano devices

Superconvergence and time evolution of discontinuous Galerkin finite element solutions