The numerical simulation of a three-dimensional semiconductor device is a fundamental problem in information science. The mathematical model is defined by an initialboundary nonlinear system of four partial differenti...The numerical simulation of a three-dimensional semiconductor device is a fundamental problem in information science. The mathematical model is defined by an initialboundary nonlinear system of four partial differential equations: an elliptic equation for electric potential, two convection-diffusion equations for electron concentration and hole concentration, and a heat conduction equation for temperature. The first equation is solved by the conservative block-centered method. The concentrations and temperature are computed by the block-centered upwind difference method on a changing mesh, where the block-centered method and upwind approximation are used to discretize the diffusion and convection, respectively. The computations on a changing mesh show very well the local special properties nearby the P-N junction. The upwind scheme is applied to approximate the convection, and numerical dispersion and nonphysical oscillation are avoided. The block-centered difference computes concentrations, temperature, and their adjoint vector functions simultaneously.The local conservation of mass, an important rule in the numerical simulation of a semiconductor device, is preserved during the computations. An optimal order convergence is obtained. Numerical examples are provided to show efficiency and application.展开更多
Numerical simulation of a three-dimensional semiconductor device is a fundamental problem in information science.The mathematical model is defined by a nonlinear system of initial-boundary problem including four parti...Numerical simulation of a three-dimensional semiconductor device is a fundamental problem in information science.The mathematical model is defined by a nonlinear system of initial-boundary problem including four partial differential equations:an elliptic equation for electrostatic potential,two convection-diffusion equations for electron concentration and hole concentration,a heat conduction equation for temperature.The electrostatic potential appears within the concentration equations and heat conduction equation,and the electric field strength controls the concentrations and the temperature.The electric field potential is solved by the conservative block-centered method,and the order of the accuracy is improved by the electric potential.The concentrations and temperature are computed by the upwind blockcentered multistep method,where three different numerical methods are involved.The multistep method is adopted to approximate the time derivative.The blockcentered method is used to discretize the diffusion.The upwind scheme is applied to approximate the convection to avoid numerical dispersion and nonphysical oscillation.The block-centered difference simulates diffusion,concentrations,temperature,and the adjoint vector functions simultaneously.It has the local conservation of mass,which is an important nature in numerical simulation of a semiconductor device.By using the variation,energy estimates,induction hypothesis,embedding theorem and the technique of a priori estimates of differential equations,convergence of the optimal order is obtained.Numerical examples are provided to show the effectiveness and viability.This method provides a powerful tool for solving the challenging benchmark problem.展开更多
In this paper,a two-grid block-centered finite difference method for the incompressible miscible displacement in porous medium is introduced and analyzed,which is to solve a nonlinear equation on coarse mesh space of ...In this paper,a two-grid block-centered finite difference method for the incompressible miscible displacement in porous medium is introduced and analyzed,which is to solve a nonlinear equation on coarse mesh space of size H and a linear equation on fine grid of size h.We establish the full discrete two-grid block-centered finite difference scheme on a uniform grid.The error estimates for the pressure,Darcy velocity,concentration variables are derived,which show that the discrete L2 error is O(Dt+h2+H4).Finally,two numerical examples are provided to demonstrate the effectiveness and accuracy of our algorithm.展开更多
In this article, two block-centered finite difference schemes are introduced and analyzed to solve the parabolic integro-differential equation arising in modeling non-Fickian flow in porous media. One scheme is Euler ...In this article, two block-centered finite difference schemes are introduced and analyzed to solve the parabolic integro-differential equation arising in modeling non-Fickian flow in porous media. One scheme is Euler backward scheme with first order accuracy in time increment while the other is Crank-Nicolson scheme with second order accuracy in time increment. Stability analysis and second-order error estimates in spatial meshsize for both pressure and velocity in discrete L^2 norms are established on non-uniform rectangular grid. Numerical experiments using the schemes show that the convergence rates are in agreement with the theoretical analysis.展开更多
A kind of conservative upwind method is discussed for chemical oil recovery displacement in porous media.The mathematical model is formulated by a nonlinear convection-diffusion system dependent on the pressure,Darcy ...A kind of conservative upwind method is discussed for chemical oil recovery displacement in porous media.The mathematical model is formulated by a nonlinear convection-diffusion system dependent on the pressure,Darcy velocity,concentration and saturations.The flow equation is solved by a conservative block-centered method,and the pressure and Darcy velocity are obtained at the same time.The concentration and saturations are determined by convection-dominated diffusion equations,so an upwind approximation is adopted to eliminate numerical dispersion and nonphysical oscillation.Block-centered method is conservative locally.An upwind method with block-centered difference is used for computing the concentration.The saturations of different components are solved by the method of upwind fractional step difference,and the computational work is shortened significantly by dividing a three-dimensional problem into three successive one-dimensional problems and using the method of speedup.Using the variation discussion,energy estimates,the method of duality,and the theory of a priori estimates,we complete numerical analysis.Finally,numerical tests are given for showing the computational accuracy,efficiency and practicability of our approach.展开更多
基金supported the Natural Science Foundation of Shandong Province(ZR2016AM08)Natural Science Foundation of Hunan Province(2018JJ2028)National Natural Science Foundation of China(11871312).
文摘The numerical simulation of a three-dimensional semiconductor device is a fundamental problem in information science. The mathematical model is defined by an initialboundary nonlinear system of four partial differential equations: an elliptic equation for electric potential, two convection-diffusion equations for electron concentration and hole concentration, and a heat conduction equation for temperature. The first equation is solved by the conservative block-centered method. The concentrations and temperature are computed by the block-centered upwind difference method on a changing mesh, where the block-centered method and upwind approximation are used to discretize the diffusion and convection, respectively. The computations on a changing mesh show very well the local special properties nearby the P-N junction. The upwind scheme is applied to approximate the convection, and numerical dispersion and nonphysical oscillation are avoided. The block-centered difference computes concentrations, temperature, and their adjoint vector functions simultaneously.The local conservation of mass, an important rule in the numerical simulation of a semiconductor device, is preserved during the computations. An optimal order convergence is obtained. Numerical examples are provided to show efficiency and application.
基金supported by National Natural Science Foundation of China(Grant No.11871312)Natural Science Foundation of Shandong Province(Grant No.ZR2021MA019).
文摘Numerical simulation of a three-dimensional semiconductor device is a fundamental problem in information science.The mathematical model is defined by a nonlinear system of initial-boundary problem including four partial differential equations:an elliptic equation for electrostatic potential,two convection-diffusion equations for electron concentration and hole concentration,a heat conduction equation for temperature.The electrostatic potential appears within the concentration equations and heat conduction equation,and the electric field strength controls the concentrations and the temperature.The electric field potential is solved by the conservative block-centered method,and the order of the accuracy is improved by the electric potential.The concentrations and temperature are computed by the upwind blockcentered multistep method,where three different numerical methods are involved.The multistep method is adopted to approximate the time derivative.The blockcentered method is used to discretize the diffusion.The upwind scheme is applied to approximate the convection to avoid numerical dispersion and nonphysical oscillation.The block-centered difference simulates diffusion,concentrations,temperature,and the adjoint vector functions simultaneously.It has the local conservation of mass,which is an important nature in numerical simulation of a semiconductor device.By using the variation,energy estimates,induction hypothesis,embedding theorem and the technique of a priori estimates of differential equations,convergence of the optimal order is obtained.Numerical examples are provided to show the effectiveness and viability.This method provides a powerful tool for solving the challenging benchmark problem.
基金National Natural Science Foundation of China No.12131014.
文摘In this paper,a two-grid block-centered finite difference method for the incompressible miscible displacement in porous medium is introduced and analyzed,which is to solve a nonlinear equation on coarse mesh space of size H and a linear equation on fine grid of size h.We establish the full discrete two-grid block-centered finite difference scheme on a uniform grid.The error estimates for the pressure,Darcy velocity,concentration variables are derived,which show that the discrete L2 error is O(Dt+h2+H4).Finally,two numerical examples are provided to demonstrate the effectiveness and accuracy of our algorithm.
基金This work is supported by the National Natural Science Foundation of China Grant no. 11671233, 91330106.
文摘In this article, two block-centered finite difference schemes are introduced and analyzed to solve the parabolic integro-differential equation arising in modeling non-Fickian flow in porous media. One scheme is Euler backward scheme with first order accuracy in time increment while the other is Crank-Nicolson scheme with second order accuracy in time increment. Stability analysis and second-order error estimates in spatial meshsize for both pressure and velocity in discrete L^2 norms are established on non-uniform rectangular grid. Numerical experiments using the schemes show that the convergence rates are in agreement with the theoretical analysis.
基金the Natural Science Foundation of Shandong Province(Grant No.ZR2021MA019)Natural Science Foundation of Hunan Province(Grant No.2018JJ2028)National Natural Science Foundation of China(Grant No.11871312).
文摘A kind of conservative upwind method is discussed for chemical oil recovery displacement in porous media.The mathematical model is formulated by a nonlinear convection-diffusion system dependent on the pressure,Darcy velocity,concentration and saturations.The flow equation is solved by a conservative block-centered method,and the pressure and Darcy velocity are obtained at the same time.The concentration and saturations are determined by convection-dominated diffusion equations,so an upwind approximation is adopted to eliminate numerical dispersion and nonphysical oscillation.Block-centered method is conservative locally.An upwind method with block-centered difference is used for computing the concentration.The saturations of different components are solved by the method of upwind fractional step difference,and the computational work is shortened significantly by dividing a three-dimensional problem into three successive one-dimensional problems and using the method of speedup.Using the variation discussion,energy estimates,the method of duality,and the theory of a priori estimates,we complete numerical analysis.Finally,numerical tests are given for showing the computational accuracy,efficiency and practicability of our approach.
