AWeighted Average Finite Difference Scheme for the Numerical Solution of Stochastic Parabolic Partial Differential Equations

In the present paper,the numerical solution of It?type stochastic parabolic equation with a timewhite noise process is imparted based on a stochastic finite difference scheme.At the beginning,an implicit stochastic finite difference scheme is presented for this equation.Some mathematical analyses of the scheme are then discussed.Lastly,to ascertain the efficacy and accuracy of the suggested technique,the numerical results are discussed and compared with the exact solution.

Optimized finite difference iterative scheme based on POD technique for 2D viscoelastic wave equation

This study develops an optimized finite difference iterative （OFDI） scheme for the two-dimensional （2D） viscoelastic wave equation. The OFDI scheme is obtained using a proper orthogonal decomposition （POD） method. It has sufficiently high accuracy with very few unknowns for the 2D viscoelastic wave equation. Existence, stability, and convergence of the OFDI solutions are analyzed. Numerical simulations verify efficiency and feasibility of the proposed scheme.

DIFFERENCE SCHEME AND NUMERICAL SIMULATION BASED ON MIXED FINITE ELEMENT METHOD FOR NATURAL CONVECTION PROBLEM

The non_stationary natural convection problem is studied. A lowest order finite difference scheme based on mixed finite element method for non_stationary natural convection problem, by the spatial variations discreted with finite element method and time with finite difference scheme was derived, where the numerical solution of velocity, pressure, and temperature can be found together, and a numerical example to simulate the close square cavity is given, which is of practical importance.

Reduced finite difference scheme and error estimates based on POD method for non-stationary Stokes equation

The proper orthogonal decomposition （POD） is a model reduction technique for the simulation Of physical processes governed by partial differential equations （e.g., fluid flows）. It has been successfully used in the reduced-order modeling of complex systems. In this paper, the applications of the POD method are extended, i.e., the POD method is applied to a classical finite difference （FD） scheme for the non-stationary Stokes equation with a real practical applied background. A reduced FD scheme is established with lower dimensions and sufficiently high accuracy, and the error estimates are provided between the reduced and the classical FD solutions. Some numerical examples illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced FD scheme based on the POD method is feasible and efficient in solving the FD scheme for the non-stationary Stokes equation.

Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

In this paper,two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation.Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense,the new schemes are proved to per-fectly preserve the total energy in the discrete sense.By using the standard energy method and the cut-off function technique,the optimal error estimates of the numerical solutions are established,and the convergence rates are of O(h^(4)+τ^(2))with mesh-size h and time-step τ.In order to improve the computational efficiency,an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step.The convergence of the iterative algorithm is also rigorously analyzed.Several numerical results are carried out to test the error estimates and conservative properties.

LARGE TIME STEP GENERALIZATION OF RANDOM CHOICE FINITE DIFFERENCE SCHEME FOR HYPERBOLIC CONSERVATION LAWS

A natural generalization of random choice finite difference scheme of Harten and Lax for Courant number larger than 1 is obtained. We handle interactions between neighboring Riemann solvers by linear superposition of their conserved quantities. We show consistency of the scheme for arbitrarily large Courant numbers. For scalar problems the scheme is total variation diminishing.A brief discussion is given for entropy condition.

THE REMAINDER－EFFECT ANALYSIS OF FINITE DIFFERENCE SCHEMES AND THE APPLICATIONS

In the present paper two contents are enclosed .First ,the Fourier analysis approach of the dispersion relation and group velocity effect of finite difference schemes is discussed.the defects of the approach is pointed out and the correction is made;Second,a new systematic analysis method -remaider -effect analysis (abbr.REAM)is proposed by means of the modified partial differential equations (abbr MPDE)of finite difference schemes.The analysis is based on the synthetical study of the rational dispersion-and dissipation relations of finite difference schemes.And the method clearly possesses constructivity

A Finite Difference Method and Effective Modification of Gradient Descent Optimization Algorithm for MHD Fluid Flow Over a Linearly Stretching Surface

Present contribution is concerned with the construction and application of a numerical method for the fluid flow problem over a linearly stretching surface with the modification of standard Gradient descent Algorithm to solve the resulted difference equation.The flow problem is constructed using continuity,and Navier Stoke equations and these PDEs are further converted into boundary value problem by applying suitable similarity transformations.A central finite difference method is proposed that gives third-order accuracy using three grid points.The stability conditions of the present proposed method using a Gauss-Seidel iterative procedure is found using Von-Neumann stability criteria and order of the finite difference method is proved by applying the Taylor series on the discretised equation.The comparison of the presently modified optimisation algorithm with the Gauss-Seidel iterative method and standard Newton’s method in optimisation is also made.It can be concluded that the presently modified optimisation Algorithm takes a few iterations to converge with a small value of the parameter contained in it compared with the standard descent algorithm that may take millions of iterations to converge.The present modification of the steepest descent method converges faster than Gauss-Seidel method and standard steepest descent method,and it may also overcome the deficiency of singular hessian arise in Newton’s method for some of the cases that may arise in optimisation problem(s).

THE UPWIND OPERATOR SPLITTING FINITE DIFFERENCE METHOD FOR COMPRESSIBLE TWO-PHASE DISPLACEMENT PROBLEM AND ANALYSIS

For compressible two-phase displacement problem, a kind of upwind operator splitting finite difference schemes is put forward and make use of operator splitting, of calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in L 2 norm are derived to determine the error, in the approximate solution.

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.

Numerical study of convective heat transfer in static arrangements of particles with arbitrary shapes:A monolithic hybrid lattice Boltzmann-finite difference-phase field solver

A compressible lattice Boltzmann-finite difference method is extended by the phase-field approach into a monolithic scheme to study fluid flow and heat transfer through regular arrangements of solid bodies of circular,elliptical and irregular shapes.The advantage of using the phase-field method is demon-strated both in its simplicity of accounting for flow and thermal boundary conditions at solid surfaces with irregular shapes and in the capability of generating such complex-shaped objects.For an array of discs,numerical results for the overall solid-to-gas heat transfer rate are validated via experiments on flow through arrays of hot cylinders.The thus validated compressible LB-FD-PF hybrid scheme is used to study the dependence of heat transfer on flow and thermal boundary conditions(Reynolds number,temperature difference between the hot solid bodies and the inlet gas),porosity as well as on the shape of solid objects.Results are rationalized in terms of the residence time of the gas close to the solid body and downstream variations of gas velocity and temperature.Perspective for further applications of the proposed methodology are also discussed.

Error Estimate of a New Conservative Finite Difference Scheme for the Klein-Gordon-Dirac System

In this paper,we derive and analyze a conservative Crank-Nicolson-type finite difference scheme for the Klein-Gordon-Dirac(KGD)system.Differing from the derivation of the existing numerical methods given in literature where the numerical schemes are proposed by directly discretizing the KGD system,we translate the KGD equations into an equivalent system by introducing an auxiliary function,then derive a nonlinear Crank-Nicolson-type finite difference scheme for solving the equivalent system.The scheme perfectly inherits the mass and energy conservative properties possessed by the KGD,while the energy preserved by the existing conservative numerical schemes expressed by two-level’s solution at each time step.By using energy method together with the‘cut-off’function technique,we establish the optimal error estimate of the numerical solution,and the convergence rate is O(τ^(2)+h^(2))in l∞-norm with time stepτand mesh size h.Numerical experiments are carried out to support our theoretical conclusions.

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

Numerical Solution of Advection Diffusion Equation Using Semi-Discretization Scheme

Numerical diffusion and oscillatory behavior characteristics are averted applying numerical solutions of advection-diffusion equation are themselves immensely sophisticated. In this paper, two numerical methods have been used to solve the advection diffusion equation. We use an explicit finite difference scheme for the advection diffusion equation and semi-discretization on the spatial variable for advection-diffusion equation yields a system of ordinary differential equations solved by Euler’s method. Numerical assessment has been executed with specified initial and boundary conditions, for which the exact solution is known. We compare the solutions of the advection diffusion equation as well as error analysis for both schemes.

High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics

This paper develops the high-order accurate entropy stable finite difference schemes for one-and two-dimensional special relativistic hydrodynamic equations.The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory(WENO)technique as well as explicit Runge-Kutta time discretization.The key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair.As soon as the entropy conservative flux is derived,the dissipation term can be added to give the semidiscrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function.The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable schemes.Several numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.

A PERTURBATIONAL FOURTH-ORDER UPWIND FINITE DIFFERENCE SCHEME FOR THE CONVECTION-DIFFUSION EQUATION

In this study,a compact fourth-order upwind finite difference scheme for the con- vection-diffusion equation is developed,by the scheme perturbation technique and the compact second-order upwind scheme proposed by the authors.The basic fourth-order scheme,which like the classical upwind scheme is free of cell Reynolds-number limitation in terms of spurious oscil- lation and involves only immediate neighbouring nodal points,is presented for the one-dimen- sional equation,and subsequently generalized to multi-dimensional cases.Numerical examples including one-to three-dimensional model equations,with available analytical solutions,of fluid flow and a problem,with benchmark solutions,of natural convective heat transfer are given to illustrate the excellent behavior in such aspects as accuracy,resolution to‘shock wave’-and ‘boundary layer’-effects in convection dominant cases,of the present scheme.Besides,the fourth-order accuracy is specially verified using double precision arithmetic.

A FINITE DIFFERENCE SCHEME FOR THE GENERALIZED NONLINEAR SCHRDINGER EQUATION WITH VARIABLE COEFFICIENTS

A finite difference scheme for the generalized nonlinear Schr dinger equation with variable coefficients is developed. The scheme is shown to satisfy two conservation laws. Numerical results show that the scheme is accurate and efficient.

Computational analysis for fractional characterization of coupled convection-diffusion equations arising in MHD fows

The work is devoted to the fractional characterization of time-dependent coupled convection-diffusion systems arising in magnetohydrodynamics(MHD)flows.The time derivative is expressed by means of Caputo’s fractional derivative concept,while the model is solved via the full-spectral method(FSM)and the semi-spectral scheme(SSS).The FSM is based on the operational matrices of derivatives constructed by using higher-order orthogonal polynomials and collocation techniques.The SSS is developed by discretizing the time variable,and the space domain is collocated by using equal points.A detailed comparative analysis is made through graphs for various parameters and tables with existing literature.The contour graphs are made to show the behaviors of the velocity and magnetic fields.The proposed methods are reasonably efficient in examining the behavior of convection-diffusion equations arising in MHD flows,and the concept may be extended for variable order models arising in MHD flows.

Effect of Viscous Dissipation (Φ) on Temperature Distribution of Blood Plasma in Presence of a Magnetic Field

Applications of heat transfer show the variations in temperature of the body which is helpful for the purpose of thermal therapy in the treatment of tumor glands. This study considered theoretical approaches in analyzing the effect of viscous dissipation on temperature distribution on the flow of blood plasma through an asymmetric arterial segment. The plasma was considered to be unsteady, laminar and an incompressible fluid through non-uniform arterial segment in a two-dimensional flow. Numerical schemes developed for the coupled partial differential equations governing blood plasma were solved using Finite Difference scheme (FDS). With the aid of the finite difference approach and the related boundary conditions, results for temperature profiles were obtained. The study determined the effect of viscous dissipation on temperature of blood plasma in arteries. The equations were solved using MATLAB softwares and results were presented graphically and in tables. The increase in viscous dissipation tends to decrease blood plasma heat distribution. This study will find important application in hospitals.

A Consistent Fourth-Order Compact Finite Difference Scheme for Solving Vorticity-Stream Function Form of Incompressible Navier-Stokes Equations

The inconsistent accuracy and truncation error in the treatment of boundary usually leads to performance defects,such as decreased accuracy and even numerical instability,of the entire computational method,especially for higher order methods.In this work,we construct a consistent fourth-order compact finite difference scheme for solving two-dimensional incompressible Navier-Stokes(N-S)equations.In the pro-posed method,the main truncation error term of the boundary scheme is kept the same as that of the interior compact finite difference scheme.With such a feature,the nu-merical stability and accuracy of the entire computation can be maintained the same as the interior compact finite difference scheme.Numerical examples show the effec-tiveness and accuracy of the present consistent compact high order scheme in L^(∞).Its application to two dimensional lid-driven cavity flow problem further exhibits that un-der the same condition,the computed solution with the present scheme is much close to the benchmark in comparison to those from the 4^(th)order explicit scheme.The compact finite difference method equipped with the present consistent boundary technique im-proves much the stability of the whole computation and shows its potential application to incompressible flow of high Reynolds number.