Mathematical Model,Numerical Simulation and Convergence Analysis of a Semiconductor Device Problem with Heat and Magnetic Influences

In this paper,the authors discuss a three-dimensional problem of the semiconductor device type involved its mathematical description,numerical simulation and theoretical analysis.Two important factors,heat and magnetic influences are involved.The mathematical model is formulated by four nonlinear partial differential equations(PDEs),determining four major physical variables.The influences of magnetic fields are supposed to be weak,and the strength is parallel to the z-axis.The elliptic equation is treated by a block-centered method,and the law of conservation is preserved.The computational accuracy is improved one order.Other equations are convection-dominated,thus are approximated by upwind block-centered differences.Upwind difference can eliminate numerical dispersion and nonphysical oscillation.The diffusion is approximated by the block-centered difference,while the convection term is treated by upwind approximation.Furthermore,the unknowns and adjoint functions are computed at the same time.These characters play important roles in numerical computations of conductor device problems.Using the theories of priori analysis such as energy estimates,the principle of duality and mathematical inductions,an optimal estimates result is obtained.Then a composite numerical method is shown for solving this problem.

An Upwind Mixed Finite Volume Element-fractional Step Method and Convergence Analysis for Three-dimensional Compressible Contamination Treatment from Nuclear Waste

In this paper the authors discuss a numerical simulation problem of three-dimensional compressible contamination treatment from nuclear waste. The mathematical model, a nonlinear convection-diffusion system of four PDEs, determines four major physical unknowns: the pressure, the concentrations of brine and radionuclide, and the temperature. The pressure is solved by a conservative mixed finite volume element method, and the computational accuracy is improved for Darcy velocity. Other unknowns are computed by a composite scheme of upwind approximation and mixed finite volume element. Numerical dispersion and nonphysical oscillation are eliminated, and the convection-dominated diffusion problems are solved well with high order computational accuracy. The mixed finite volume element is conservative locally, and get the objective functions and their adjoint vector functions simultaneously. The conservation nature is an important character in numerical simulation of underground fluid. Fractional step difference is introduced to solve the concentrations of radionuclide factors, and the computational work is shortened significantly by decomposing a three-dimensional problem into three successive one-dimensional problems. By the theory and technique of a priori estimates of differential equations, we derive an optimal order estimates in L2norm. Finally, numerical examples show the effectiveness and practicability for some actual problems.

A Mixed-finite Volume Element Coupled with the Method of Characteristic Fractional Step Difference for Simulating Transient Behavior of Semiconductor Device of Heat Conductor And Its Numerical Analysis

The mathematical system is formulated by four partial differential equations combined with initial- boundary value conditions to describe transient behavior of three-dimensional semiconductor device with heat conduction. The first equation of an elliptic type is defined with respect to the electric potential, the successive two equations of convection dominated diffusion type are given to define the electron concentration and the hole concentration, and the fourth equation of heat conductor is for the temperature. The electric potential appears in the equations of electron concentration, hole concentration and the temperature in the formation of the intensity. A mass conservative numerical approximation of the electric potential is presented by using the mixed finite volume element, and the accuracy of computation of the electric intensity is improved one order. The method of characteristic fractional step difference is applied to discretize the other three equations, where the hyperbolic terms are approximated by a difference quotient in the characteristics and the diffusion terms are discretized by the method of fractional step difference. The computation of three-dimensional problem works efficiently by dividing it into three one-dimensional subproblems and every subproblem is solved by the method of speedup in parallel. Using a pair of different grids （coarse partition and refined partition）, piecewise threefold quadratic interpolation, variation theory, multiplicative commutation rule of differential operators, mathematical induction and priori estimates theory and special technique of differential equations, we derive an optimal second order estimate in L2-norm. This numerical method is valuable in the simulation of semiconductor device theoretically and actually, and gives a powerful tool to solve the international problem presented by J. Douglas, Jr.

A Symmetric Characteristic Finite Volume Element Scheme for Nonlinear Convection-Diffusion Problems

In this paper, we implement alternating direction strategy and construct a symmetric FVE scheme for nonlinear convection-diffusion problems. Comparing to general FVE methods, our method has two advantages. First, the coefficient matrices of the discrete schemes will be symmetric even for nonlinear problems. Second, since the solution of the algebraic equations at each time step can be inverted into the solution of several one-dimensional problems, the amount of computation work is smaller. We prove the optimal H1-norm error estimates of order O（△t2 ＋ h） and present some numerical examples at the end of the paper.

Theory and Application of Characteristic Finite Element Domain Decomposition Procedures for Coupled System of Dynamics of Fluids in Porous Media

For a coupled system of multiplayer dynamics of fluids in porous media, the characteristic finite element domain decomposition procedures applicable to parallel arithmetic are put forward. Techniques such as calculus of variations, domain decomposition, characteristic method, negative norm estimate, energy method and the theory of prior estimates are adopted. Optimal order estimates in L^2 norm are derived for the error in the approximate solution.