Numerical Analysis of Upwind Difference Schemes for Two-Dimensional First-Order Hyperbolic Equations with Variable Coefficients

In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.

NUMERICAL METHOD OF MIXED FINITE VOLUME-MODIFIED UPWIND FRACTIONAL STEP DIFFERENCE FOR THREE-DIMENSIONAL SEMICONDUCTOR DEVICE TRANSIENT BEHAVIOR PROBLEMS

Transient behavior of three-dimensional semiconductor device with heat conduc- tion is described by a coupled mathematical system of four quasi-linear partial differential equations with initial-boundary value conditions. The electric potential is defined by an ellip- tic equation and it appears in the following three equations via the electric field intensity. The electron concentration and the hole concentration are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite volume element approximation, keeping physical conservation law, is used to get numerical values of the electric potential and the accuracy is improved one order. Two con- centrations and the heat conduction are computed by a fractional step method combined with second-order upwind differences. This method can overcome numerical oscillation, dispersion and decreases computational complexity. Then a three-dimensional problem is solved by computing three successive one-dimensional problems where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by using prior estimate theory and other special techniques of partial differential equations. This type of mass-conservative parallel method is important and is most valuable in numerical analysis and application of semiconductor device.

THE UPWIND FINITE DIFFERENCE METHOD FOR MOVING BOUNDARY VALUE PROBLEM OF COUPLED SYSTEM

Coupled system of multilayer dynamics of fluids in porous media is to describe the history of oil-gas transport and accumulation in basin evolution.It is of great value in rational evaluation of prospecting and exploiting oil-gas resources.The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values.The upwind finite difference schemes applicable to parallel arithmetic are put forward and two-dimensional and three-dimensional schemes are used to form a complete set.Some techniques,such as change of variables,calculus of variations, multiplicative commutation rule of difference operators,decomposition of high order difference operators and prior estimates,are adopted.The estimates in l~2 norm are derived to determine the error in the approximate solution.This method was already applied to the numerical simulation of migration-accumulation of oil resources.

Upwind finite difference method for miscible oil and water displacement problem with moving boundary values

The research of the miscible oil and water displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in the basin evolution as well as to the rational evaluation in prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. For the twodimensional bounded region, the upwind finite difference schemes are proposed. Some techniques, such as the calculus of variations, the change of variables, and the theory of a priori estimates, are used. The optimal orderl2-norm estimates are derived for the errors in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, the model numerical method, and the software development.

Comments on Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

The explicit compact difference scheme, proposed in Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD by Lin et al., published in Applied Mathematics and Mechanics （English Edition）, 2007, 28（7）, 943-953, has the same performance as the conventional finite difference schemes. It is just another expression of the conventional finite difference schemes. The proposed expression does not have the advantages of a compact difference scheme. Nonetheless, we can more easily obtain and implement compared with the conventional expression in which the coefficients can only be obtained by solving equations, especially for higher accurate schemes.

A BLOCK-CENTERED UPWIND APPROXIMATION OF THE SEMICONDUCTOR DEVICE PROBLEM ON A DYNAMICALLY CHANGING MESH

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.

Numerical simulation for 2D double-diffusive convection(DDC) in rectangular enclosures based on a high resolution upwind compact streamfunction model Ⅰ: numerical method and code validation

A high resolution upwind compact streamfunction numerical algorithm for two-dimensional(2D)double-diffusive convection(DDC)is developed.The unsteady Navier-Stokes(N-S)equations in the streamfunction-velocity form and the scalar temperature and concentration equations are used.An optimized third-order upwind compact(UCD3 opt)scheme with a low dispersion error for the first derivatives is utilized to approximate the third derivatives of the streamfunction in the advection terms of the N-S equations and the first derivatives in the advection terms of the scalar temperature and concentration equations.The remaining first derivatives of the streamfunction(velocity),temperature,and concentration variables used in the governing equations are discretized by the fourth-order compact Pade(SCD4)schemes.With the temperature and concentration variables and their approximate values of the first derivatives obtained by the SCD4 schemes,the explicit fourth-order compact schemes are suggested to approximate the second derivatives of temperature and concentration in the diffusion terms of the energy and concentration equations.The discretization of the temporal term is executed with the second-order Crank-Nicolson(C-N)scheme.To assess the spatial behavior capability of the established numerical algorithm and verify the developed computer code,the DDC flow is numerically solved.The obtained results agree well with the benchmark solutions and some accurate results available in the literature,verifying the accuracy,effectiveness,and robustness of the provided algorithm.Finally,a preliminary application of the proposed method to the DDC is carried out.

UNSTEADY/STEADY NUMERICAL SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON ARTIFICIAL COMPRESSIBILITY

A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.

Application of Mixed Differential Quadrature Method for Solving the Coupled Two-Dimensional Incompressible Navier-Stokes Equation and Heat Equation

The traditional differential quadrature method was improved by using theupwind difference scheme for the convective terms to solve the coupled two-dimensionalincompressible Navier-stokes equations and heat equation. The new method was compared with theconventional differential quadrature method in the aspects of convergence and accuracy. The resultsshow that the new method is more accurate, and has better convergence than the conventionaldifferential quadrature method for numerically computing the steady-state solution.

NUMERICAL SOLUTION OF A SINGULARLY PERTURBED ELLIPTIC-HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION ON A NONUNIFORM DISCRETIZATION MESH

In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter e , to the solution of problem (1.1). Numerical results are finally provided.

A Spectral Method for Convection-Diffusion Equations

In the practical problems such as nuclear waste pollution and seawater intrusion etc., many problems are reduced to solving the convection-diffusion equation, so the research of convection-diffusion equation is of great value. In this work, a spectral method is presented for solving one and two dimensional convection-diffusion equation with source term. The finite difference method is also used to solve the convection diffusion equation. The numerical experiments show that the spectral method is more efficient than other methods for solving the convection-diffusion equation.

GROUP VELOCITY CONTROL SCHEME WITH LOW DISSIPATION

In order to prevent smearing the discontinuity, a modified term is added to the third order Upwind Compact Difference scheme to lower the dissipation error. Moreover, the dispersion error is controled to hold back the non physical oscillation by means of the group velocity control. The scheme is used to simulate the interactions of shock density stratified interface and the disturbed interface developing to vortex rollers. Numerical results are satisfactory.

THE UPWIND FINITE DIFFERENCE FRACTIONAL STEPS METHOD FOR NONLINEAR COUPLED SYSTEM OF DYNAMICS OF FLUIDS IN POROUS MEDIA

For nonlinear coupled system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward, trod two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L2 norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources.

The Modified Upwind Finite Difference Fractional Steps Method for Compressible Two-phase Displacement Problem

For compressible two-phase displacement problem,the modified upwind finite difference fractionalsteps schemes are put forward.Some techniques,such as calculus of variations,commutative law of multiplicationof difference operators,decomposition of high order difference operators,the theory of prior estimates and tech-niques are used.Optimal order estimates in L^2 norm are derived for the error in the approximate solution.Thismethod has already been applied to the numerical simulation of seawater intrusion and migration-accumulationof oil resources.

AConservative Upwind Approximation on Block-Centered Difference for Chemical Oil Recovery Displacement Problem

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.

Upwind and High-Resolution Methods for Compressible Flow:From Donor Cell to Residual-Distribution Schemes

In this paper I review three key topics in CFD that have kept researchers busy for half a century.First,the concept of upwind differencing,evident for 1-D linear advection.Second,its implementation for nonlinear systems in the form of highresolution schemes,now regarded as classical.Third,its genuinely multidimensional implementation in the form of residual-distribution schemes,the most recent addition.This lecture focuses on historical developments;it is not intended as a technical review of methods,hence the lack of formulas and absence of figures.

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.

PARAMETER-UNIFORM FINITE DIFFERENCE SCHEME FOR A SYSTEM OF COUPLED SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATIONS

A coupled system of singularly perturbed convection-diffusion equations is considered. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solutions to the system have boundary layers that overlap and interact. The structure of these layers is analyzed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh an upwind difference scheme is proved to be almost first- order accurate, uniformly in both small parameters. We present the results of numerical experiments to confirm our theoretical results.

A Third-Order Upwind Compact Scheme on Curvilinear Meshes for the Incompressible Navier-Stokes Equations

This paper presents a new version of the upwind compact finite difference scheme for solving the incompressible Navier-Stokes equations in generalized curvilinear coordinates.The artificial compressibility approach is used,which transforms the elliptic-parabolic equations into the hyperbolic-parabolic ones so that flux difference splitting can be applied.The convective terms are approximated by a third-order upwind compact scheme implemented with flux difference splitting,and the viscous terms are approximated by a fourth-order central compact scheme.The solution algorithm used is the Beam-Warming approximate factorization scheme.Numerical solutions to benchmark problems of the steady plane Couette-Poiseuille flow,the liddriven cavity flow,and the constricting channel flow with varying geometry are presented.The computed results are found in good agreement with established analytical and numerical results.The third-order accuracy of the scheme is verified on uniform rectangular meshes.

PARALLEL IMPLEMENTATIONS OF THE FAST SWEEPING METHOD

The fast sweeping method is an efficient iterative method for hyperbolic problems. It combines Gauss-Seidel iterations with alternating sweeping orderings. In this paper several parallel implementations of the fast sweeping method are presented. These parallel algorithms are simple and efficient due to the causality of the underlying partial different equations. Numerical examples are used to verify our algorithms.