A-high-order Accuraqcy Implicit Difference Scheme for Solving the Equation of Parabolic Type

In this paper,a implicit difference scheme is proposed for solving the equation of one_dimension parabolic type by undetermined paameters.The stability condition is r=αΔt/Δx 2 1/2 and the truncation error is o(Δt 4+Δx 4) It can be easily solved by double sweeping method.

LONG TIME ASYMPTOTIC BEHAVIOR OF SOLUTION OF IMPLICIT DIFFERENCE SCHEME FOR A SEMI-LINEAR PARABOLIC EQUATION

In this paper, the solution of back-Euler implicit difference scheme for a semi-linear parabolic equation is proved to converge to the solution of difference scheme for the corresponding semi-linear elliptic equation as t tends to infinity. The long asymptotic behavior of its discrete solution is obtained which is analogous to that of its continuous solution. At last, a few results are also presented for Crank-Nicolson scheme.

IMPLICIT DIFFERENCE APPROXIMATION FOR A TIME FRACTIONAL ADVECTION-DISPERSION EQUATION

In this paper, a time fractional advection-dispersion equation is considered. From the relationship between the Caputo derivative and the Griinwald derivative, the Caputo derivative is approximated by using the Griinwald derivative. An implicit difference approximation for this equation is proposed. We prove that this approximation is unconditionally stable and convergent. Finally, numerical examples are given.

A truncated implicit high-order finite-difference scheme combined with boundary conditions

In this paper, first we calculate finite-difference coefficients of implicit finite- difference methods （IFDM） for the first and second-order derivatives on normal grids and first- order derivatives on staggered grids and find that small coefficients of high-order IFDMs exist. Dispersion analysis demonstrates that omitting these small coefficients can retain approximately the same order accuracy but greatly reduce computational costs. Then, we introduce a mirrorimage symmetric boundary condition to improve IFDMs accuracy and stability and adopt the hybrid absorbing boundary condition （ABC） to reduce unwanted reflections from the model boundary. Last, we give elastic wave modeling examples for homogeneous and heterogeneous models to demonstrate the advantages of the proposed scheme.

THE HIGH ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING PARABOLIC EQUATIONS 3-DIMENSION

In this paper, an explicit three_level symmetrical differencing scheme with parameters for solving parabolic partial differential equation of three_dimension will be considered. The stability condition and local truncation error for the scheme are r<1/2 and O( Δ t 2+ Δ x 4+ Δ y 4+ Δ z 4) ,respectively.

Effects of Prandtl and Jacob Numbers and Dimensionless Thermal Conductivity on Velocity Profiles in Media (Porous and Liquid)

In this present study, we analyzed the effects of Prandtl and Jacob numbers and dimensionless thermal conductivity on the velocity profiles in media (porous and liquid). The transfers in the porous medium and the liquid film are described respectively by the improved Wooding model and the classical boundary layer equations. The mesh of the digital domain is considered uniform in the transverse and longitudinal directions. The advection and diffusion terms are discretized with a back-centered and centered scheme respectively. The coupled systems of algebraic equations thus obtained are solved numerically using an iterative line-by-line relaxation method of the Gauss-Seidel type. The results show that the parameters relating to the thermal problem (the dimensionless thermal conductivity, the Prandtl (Pr) and Jacob (Ja) numbers) have no influence on the dimensionless speed, although the thermal and hydrodynamic problems are coupled. Via the heat balance equation. The results obtained show that the parameters relating to the thermal problem have no influence on the dimensionless speed, although the thermal and hydrodynamic problems are coupled via the heat balance equation. So, at first approximation with the chosen constants, we can solve the hydrodynamic problem independently of the thermal problem.

Magnetohydrodynamic Conjugate Free Convective Heat Transfer Analysis of an Isothermal Horizontal Circular Cylinder with Temperature Dependent Viscosity

Heat transfers due to MHD-conjugate free convection from the isothermal horizontal circular cylinder while viscosity is a function of temperature is investigated. The governing equations of the flow and connected boundary conditions are made dimensionless using a set of non-dimensional parameters. The governing equations are solved numerically using the finite difference method. Numerical results are obtained for various values of viscosity variation parameter, Prandtl number, magnetic parameter, and conjugate conduction parameter for the velocity and the temperature within the boundary layer as well as the skin friction coefficients and heat transfer rate along the surface.

THEORY AND APPLICATION OF FRACTIONAL STEP CHARACTERISTIC FINITE DIFFERENCE METHOD IN NUMERICAL SIMULATION OF SECOND ORDER ENHANCED OIL PRODUCTION

A kind of second-order implicit fractional step characteristic finite difference method is presented in this paper for the numerically simulation coupled system of enhanced （chemical） oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in l^2 norm is derived. This method has been applied successfully to the numerical simulation of enhanced oil production in actual oilfields, and the simulation results ate quite interesting and satisfactory.

New algorithm for solving 3D incompressible viscous equations based on projection method

A new algorithm based on the projection method with the implicit finite difference technique was established to calculate the velocity fields and pressure.The calculation region can be divided into different regions according to Reynolds number.In the far-wall region,the thermal melt flow was calculated as Newtonian flow.In the near-wall region,the thermal melt flow was calculated as non-Newtonian flow.It was proved that the new algorithm based on the projection method with the implicit technique was correct through nonparametric statistics method and experiment.The simulation results show that the new algorithm based on the projection method with the implicit technique calculates more quickly than the solution algorithm-volume of fluid method using the explicit difference method.

High-order numerical methods of fractional-order Stokes' first problem for heated generalized second grade fluid

The high-order implicit finite difference schemes for solving the fractional- order Stokes＇ first problem for a heated generalized second grade fluid with the Dirichlet boundary condition and the initial condition are given. The stability, solvability, and convergence of the numerical scheme are discussed via the Fourier analysis and the matrix analysis methods. An improved implicit scheme is also obtained. Finally, two numerical examples are given to demonstrate the effectiveness of the mentioned schemes

Simulation on venting process and valve opening control method for gas trunk pipelines

Determining the venting time of a gas trunk pipeline segment provides an important basis for formulating an emergency plan in the advent of unexpected accidents.As the natural gas venting process corresponds to the transient flow,it is necessary to establish a transient hydraulic-thermal simulation model in order to determine the venting time.In this paper,based on two kinds of venting scenarios in which there is only one venting point in the venting system of a gas trunk pipeline segment—namely,where the venting point is either at one of the two ends or at the junction of two gas trunk pipeline segments—transient hydraulic-thermal simulation models are established.The models consist of gas flow governing equations,the gas state equation,gas physical property equations,initial conditions,and appropriate boundary conditions.The implicit central difference method is used to discretize the gas flow partial differential equations,and the trust-region-dogleg algorithm is used to solve the equations corresponding to each time step,in order to dynamically simulate the whole venting process.The judgment condition for the end of the venting process is that the average pressure of gas trunk pipeline segment is less than 0.11 MPa(actual pressure).Comparing the simulation results of the proposed model with those of the OLGA software and real operational data,we find that the venting time error is less than 10%.On this basis,a venting valve opening control principle is proposed,which prevents the venting noise from exceeded the specified noise value(85 d B)in the venting design of domestic gas pipeline projects.The established calculation model for venting time(dynamic simulation model)for a gas trunk pipeline segment and the proposed opening control principle of venting valve provide reference for the optimal operation of gas pipeline venting systems.

Reliable approach for bistatic scattering of three-dimensional targets from underlying rough surface based on parabolic equation

A parabolic equation （PE） based method for analyzing composite scattering under an electromagnetic wave incidence at low grazing angle, which composes of three-dimensional （3-D） electrically large targets and rough surface, is presented and discussed. A superior high-order PE version is used to improve the accuracy at wider paraxial angles, and along with the alternating direction implicit （ADI） differential technique, the computational efficiency is further improved. The formula of bistatic normalized radar cross section is derived by definition and near-far field transformation. Numerical examples are given to show the validity and accuracy of the proposed approach, in which the results are compared with those of Kirchhoff approximation （KA） and moment of method （MoM）. Furthermore, the bistatic scattering properties of composite model in which the 3-D PEC targets on or above the two-dimensional Gaussian rough surfaces under the tapered wave incidence are analyzed.

Numerical Study of Thin Film Condensation in Forced Convection of Saturated Vapor on a Vertical Wall Covered with a Porous Material

The study of forced convection in a porous medium has aroused and still arouses today the interest of many scientists and industrialists. A considerable amount of work has been undertaken following the discovery of the phenomenon. Solving a standard problem of forced convection in porous media comes down to predicting the temperature and velocity fields as well as the intensity of the flow as a function of the various parameters of the problem. A numerical study of the condensation in forced convection of a pure and saturated vapor on a vertical wall covered with a porous material is presented. The transfers in the porous medium and the liquid film are described respectively by the Darcy-Brinkman model and the classical boundary layer equations. The dimensionless equations are solved by an implicit finite difference method and the iterative Gauss-Seidel method. Our study makes it possible to examine and highlight the role of parameters such as: the Froude number and the thickness of the porous layer on the speed and the temperature in the porous medium. Given the objective of our study, the presentation of velocity and temperature profiles is limited in the porous medium. The results show that the Froude number does not influence the thermal field. The temperature increases with an increase in the thickness of the dimensionless porous layer. The decrease in the Froude number leads to an increase in the hydrodynamic field.

Numerical Study of Thin Film Condensation in Forced Convection on an Inclined Wall Covered with a Porous Material

The present work presents a study of forced convection condensation of a laminar film of a pure and saturated vapor on a porous plate inclined to the vertical. The Darcy-Brinkman-Forchheimer model is used to write the flow in the porous medium, while the classical boundary layer equations have been exploited in the pure liquid and in the porous medium taking into account inertia and enthalpy convection terms. The problem has been solved numerically. The results are mainly presented in the form of velocity and temperature profiles. The obtained results have been compared with the numerical results of Chaynane et al. [1]. The effects of different influential parameters such as: inclination (ϕ), effective viscosity (ReK), and dimensionless thermal conductivity λ* on the flow and heat transfers are outlined.

LONG TIME ASYMPTOTIC BEHAVIOR OF SOLUTION OF DIFFERENCE SCHEME FOR A SEMILINEAR PARABOLIC EQUATION (I)

In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as t -&gt infinity. For the discrete solution of nonlinear parabolic problem, we get its long time asymptotic behavior which is similar to that of the continuous solution. For simplicity, we consider one-dimensional problem.

NUMERICAL EXPERIMENTS ABOUT GROWTH OF THE ENERGY OF DISTURBANCE IN A BAROCLINIC FLOW

This article is concerned with the growth of energy of disturbances in a baroclinic flow within a finite time period. The implicit difference scheme was applied to the linearized vorticity equation, and the disturbance energy was computed for three kinds of vertical shears. It turns out that all the disturbance energy rapidly increases initially, and during the succeeding period there are several stages of growth and decay of energy of disturbances, and from a certain time on, all the disturbance energy begins to decrease.

Numerical solution of micropolar fluid flow via stretchable surface with chemical reaction and melting heat transfer using Keller-Box method

The main theme of this research is to find the numerical results of stagnation point flow of micropolar fluid over a porous stretchable surface due to the physical effects of internal heat generation/absorption,melting heat transfer and chemical reaction via Keller-Box method(KBM).The graphs and tables are depicted and explained for various embedded parameters.The range of melting heat transfer parameter is 0≤M≤3,the range of chemical reaction parameter is 0≤K_(r)≤1 whereas the values of space-temperature dependent heat source/sink parameters lies in-0:4≤Q≤0:4 and-2≤Q*≤2.The upshots of the current problem illustrate that at fluid-solid interface,rate of HMT(heat and mass transfer)declined on escalating the values of stretching parameter.Moreover,as the values of internal heat source/sink parameter increases,heat transfer rate declines at fluid-solid interface.

NUMERICAL SOLUTION OF THE SPACE FRACTIONAL DIFFERENTIAL EQUATION

In this paper, a space fractional differential equation is considered. The equation is obtained from the parabolic equation containing advection, diffusion and reaction terms by replacing the second order derivative in space by a fractional derivative in space of order. An implicit finite difference approximation for this equation is presented. The stability and convergence of the finite difference approximation are proved. A fractional-order method of lines is also presented. Finally, some numerical results are given.

A MFE method combined with L1-approximation for a nonlinear time-fractional coupled diffusion system

In this paper,a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element(MFE)method in space combined with L1-approximation and implicit second-order backward difference scheme in time.The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived.Finally,some numerical tests are shown to verify our theoretical analysis.