A CLASS OF EXPLICIT FORWARD TIME-DIFFERENCE SQUARE CONSERVATIVE SCHEMES

In the present paper, a class of explicit forward time-difference schemes are established from a geometric view with strict analytical deductions. This class includes the schemes with a constant time interval and with adjustable time intervals, which is proved to be effective and remarkably time-saving in numerical tests and applications.

A CLASS OF TWO-LEVEL EXPLICIT DIFFERENCE SCHEMES FOR SOLVING THREE DIMENSIONAL HEAT CONDUCTION EQUATION

A class of two-level explicit difference schemes are presented for solving three-dimensional heat conduction equation. When the order of truncation error is 0(Deltat + (Deltax)(2)), the stability condition is mesh ratio r = Deltat/(Deltax)(2) = Deltat/(Deltay)(2) = Deltat/(Deltaz)(2) less than or equal to 1/2, which is better than that of all the other explicit difference schemes. And when the order of truncation error is 0((Deltat)(2) + (Deltax)(4)), the stability condition is r less than or equal to 1/6, which contains the known results.

A FAMILY OF HIGH-ORDER ACCURACY EXPLICIT DIFFERENCE SCHEMES WITH BRANCHING STABILITY FOR SOLVING 3-D PARABOLIC PARTIAL DIFFERENTIAL EQUATION

A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).

EXPLICIT SQUARE CONSERVING SCHEMES OF LANDAU-LIFSHITZ EQUATIONWITH GILBERT COMPONENT

A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equations. Then the Lie group method and the Runge-Kutta (RK) method were applied to the ordinary differential equations. The square conserving property and the accuracy of the two methods were compared. Numerical experiment results show the Lie group method has the good accuracy and the square conserving property than the RK method.

Constructions and Applied Examinations of a Kind of Square-Conservative Schemes in High Precision in the Time Direction

In order to meet the needs of work in numerical weather forecast and in numerical simulations for climate change and ocean current, a kind of difference scheme in high precision in the time direction developed from the completely square-conservative difference scheme in explicit way is built by means of the Taylor expansion. A numerical test with 4-wave Rossby-Haurwitz waves on them and an application of them on the monthly mean current the of South China Sea are carried out, from which, it is found that not only do the new schemes have high harmony and approximate precision but also can the time step of the schemes be lengthened and can much computational time be saved. Therefore, they are worth generalizing and applying.

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.

A Compact Explicit Difference Scheme of High Accuracy for Extended Boussinesq Equations

Presented here is a compact explicit difference scheme of high accuracy for solving the extended Boussinesq equations. For time discretization, a three-stage explicit Runge-Kutta method with TVD property is used at predicting stage, a cubic spline function is adopted at correcting stage, which made the time discretization accuracy up to fourth order; For spatial discretization, a three-point explicit compact difference scheme with arbitrary order accuracy is employed. The extended Boussinesq equations derived by Beji and Nadaoka are solved by the proposed scheme. The numerical results agree well with the experimental data. At the same time, the comparisons of the two numerical results between the present scheme and low accuracy difference method are made, which further show the necessity of using high accuracy scheme to solve the extended Boussinesq equations. As a valid sample, the wave propagation on the rectangular step is formulated by the present scheme, the modelled results are in better agreement with the experimental data than those of Kittitanasuan.

A－HIGH－ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THE EQUATION OF TWO－DIMENSIONAL PARABOLIC TYPE

In this paper. a three explicit difference shcemes with high order accuracy for solving the equations of two-dimensional parabolic type is proposed. The stability condition is r=△t/△x￣ 2=△t/△y￣2≤1/4 and the truncation error is O (△t￣2 + △x￣4 ).

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.

Axisymmetric alternating direction explicit scheme for efficient coupled simulation of hydro-mechanical interaction in geotechnical engineering-Application to circular footing and deep tunnel in saturated ground

Explicit solution techniques have been widely used in geotechnical engineering for simulating the coupled hydro-mechanical(H-M) interaction of fluid flow and deformation induced by structures built above and under saturated ground, i.e. circular footing and deep tunnel. However, the technique is only conditionally stable and requires small time steps, portending its inefficiency for simulating large-scale H-M problems. To improve its efficiency, the unconditionally stable alternating direction explicit(ADE)scheme could be used to solve the flow problem. The standard ADE scheme, however, is only moderately accurate and is restricted to uniform grids and plane strain flow conditions. This paper aims to remove these drawbacks by developing a novel high-order ADE scheme capable of solving flow problems in nonuniform grids and under axisymmetric conditions. The new scheme is derived by performing a fourthorder finite difference(FD) approximation to the spatial derivatives of the axisymmetric fluid-diffusion equation in a non-uniform grid configuration. The implicit Crank-Nicolson technique is then applied to the resulting approximation, and the subsequent equation is split into two alternating direction sweeps,giving rise to a new axisymmetric ADE scheme. The pore pressure solutions from the new scheme are then sequentially coupled with an existing geomechanical simulator in the computer code fast Lagrangian analysis of continua(FLAC). This coupling procedure is called the sequentially-explicit coupling technique based on the fourth-order axisymmetric ADE scheme or SEA-4-AXI. Application of SEA-4-AXI for solving axisymmetric consolidation of a circular footing and of advancing tunnel in deep saturated ground shows that SEA-4-AXI reduces computer runtime up to 42%-50% that of FLAC’s basic scheme without numerical instability. In addition, it produces high numerical accuracy of the H-M solutions with average percentage difference of only 0.5%-1.8%.

A NEW HIGH-ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THREE-DIMENSIONAL PARABOLIC EQUATIONS

In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gamma(2) = Delta t/Delta z(2) less than or equal to 1/4, and the truncation error is O(Delta t(2) + Delta x(4)).

Efficient Finite Difference Methods for the Numerical Analysis of One-Dimensional Heat Equation

In this paper, we investigate and analyze one-dimensional heat equation with appropriate initial and boundary condition using finite difference method. Finite difference method is a well-known numerical technique for obtaining the approximate solutions of an initial boundary value problem. We develop Forward Time Centered Space (FTCS) and Crank-Nicolson (CN) finite difference schemes for one-dimensional heat equation using the Taylor series. Later, we use these schemes to solve our governing equation. The stability criterion is discussed, and the stability conditions for both schemes are verified. We exhibit the results and then compare the results between the exact and approximate solutions. Finally, we estimate error between the exact and approximate solutions for a specific numerical problem to present the convergence of the numerical schemes, and demonstrate the resulting error in graphical representation.

Chloride Ion Transmission Model under the Drying-wetting Cycles and Its Solution

The chloride ion transmission model considering diffusion and convection was established respectively for different zones in concrete by analyzing chloride ion transmission mechanism under the dryingwetting cycles. The finite difference method was adopted to solve the model. The equation of chloride ion transmission model in the convection and diffusion zone of concrete was discreted by the group explicit scheme with right single point （GER method） and the equation in diffusion zone was discreted by FTCS difference scheme. According to relative humidity characteristics in concrete under drying-wetting cycles, the seepage velocity equation was formulated based on Kelvin Equation and Darcy＇s Law. The time-variant equations of chloride ion concentration of concrete surface and the boundary surface of the convection and diffusion zone were established. Based on the software MATLAB the numerical calculation was carried out by using the model and basic material parameters from the experiments. The calculation of chloride ion concentration distribution in concrete is in good agreement with the drying-wetting cycles experiments. It can be shown that the chloride ion transmission model and the seepage velocity equation are reasonable and practical. Studies have shown that the chloride ion transmission in concrete considering convection and diffusion under the drying-wetting cycles is the better correlation with the actual situation than that only considering the diffusion.

An Improved Splitting Method

In this paper, an improved splitting method, based on the completely square-conservative explicit difference schemes, is established. Not only can the time-direction precision of this method be higher than that of the traditional splitting methods but also can the physical feature of mutual dependence of the fast and the slow stages that are calculated separately and splittingly be kept as well. Moreover, the method owns an universality, it can be generalized to other square-conservative difference schemes, such as the implicit and complete ones and the explicit and instantaneous ones. Good time benefits can be acquired when it is applied in the numerical simulations of the monthly mean currents of the South China Sea.

EXPLICIT DISSIPATIVE SCHEMES FOR BOUNDARY PROBLEMS OF GENERALIZED SCHRDINGER SYSTEMS

In this paper, the author constructs a class of explicit schemes, spanning two time levels, forthe initial--boundary-value problems of generalized nonlinear Schrodinger systems, and proves theconvergence of these schemes with a series of prior estimates. For a single Schrodinger equation, theschemes are identical with those of the article [1].

Numerical Methods for Solving Logarithmic Nonlinear Schrödinger’s Equation

In this study, we will construct numerical techniques for tackling the logarithmic Schr&#246;dinger’s nonlinear equation utilizing the explicit scheme and the Crank-Nicolson scheme of the finite difference method. These schemes will be subjected to accuracy and stability tests before being used. Efficacy and robustness of the techniques under consideration will be demonstrated using an exact solution, one-Gausson, as well as conserved quantities. Interaction of two-soliton will be conducted. The numerical findings revealed, the interplay behavior is flexible.

A Numerical Solution of Heat Equation for Several Thermal Diffusivity Using Finite Difference Scheme with Stability Conditions

The heat equation is a second-order parabolic partial differential equation, which can be solved in many ways using numerical methods. This paper provides a numerical solution that uses the finite difference method like the explicit center difference method. The forward time and centered space (FTCS) is used to a problem containing the one-dimensional heat equation and the stability condition of the scheme is reported with different thermal conductivity of different materials. In this study, results obtained for different thermal conductivity of distinct materials are compared. Also, the results reveal the well-behavior properties of the materials in good agreement.

Simulation of open channel flows by an explicit incompressible mesh-free method

In this study,to simulate open channel flows,an explicit incompressible mesh-free method is employed in which the pressure field is obtained by explicitly solving the pressure Poisson equation.To capture the velocity information in open channel flows,the source term in the pressure Poisson equation is modified while the spatial discretization of gradient and Laplacian models is based on the moving particle semi-implicit(MPS)method.The inflow boundary condition is treated by injecting fluid particles into the domain according to the inlet discharge,and the outflow condition is handled by prescribing the pressure distribution and removing the fluid particles beyond the domain.The explicit incompressible mesh-free method is then used to simulate open channel flows,including weir flow,hydraulic jump,and flow over an obstacle.In the simulations,velocity distribution and flow pattern are examined.The simulated results are compared to available experimental measurements and other numerical results.There is a good agreement between the simulated results and the experimental measurements.It shows that the explicit incompressible mesh-free method can reproduce the flow characteristics in the open channel flows.

REDUCED AUGMENTED LAGRANGIAN BI-CONJUGATE GRADIENT METHOD FOR IMPACT-CONTACT PROBLEMS

To avoid the numerical oscillation of the penalty method andnon-compatibility with ex- plicit operators of conventional Lagrangemultiplier methods used in transient contact problems to en- forcesurface contact conditions, a new approach to enforcing surfacecontact constraints for the tran- sient nonlinear finite elementproblems, referred to as 'the reduced augmented Lagrangianbi-conjugate gradient method (ALCG)', is developed in this paper.Based on the nonlinear constrained optimization theory and iscompatible with the explicit time integration scheme, this approachcan also be used in implicit scheme naturally.

AN EXPLICIT COMPLETE SQUARE CONSERVATIVE SCHEME WITH ADJUSTABLE TIME INTERVALS

In this paper,a kind of explicit difference scheme to solve nonlinear evolution equations,perfectly keeping the square conservation by adjusting the time step interval,is constructed,from the comprehensive maintenance of the ad- vantages of the implicit complete square conservative scheme and the explicit instantaneous square conservative scheme. The new schemes are based on the thought of adding a small dissipation,but it is different from the small dissipation method.The dissipative term used in the new schemes is not a simple artificial dissipative term,but a so-called (time) harmonious dissipative term that can compensate for the truncation errors from the dissociation of the time differential term.Therefore,the new schemes may have a high time precision and may acquire a satisfactory effect in numerical tests.