A Family of High_order Accuracy Explicit Difference Schemes for Solving 2-D Parabolic Partial Differential Equation

A family of high_order accuracy explicit difference schemes for solving 2_dimension parabolic P.D.E. are constructed. Th e stability condition is r=Δt/Δx 2=Δt/Δy 2【1/2 and the truncation err or is O(Δt 3+Δx 4).

Parallel EOI algorithm with different insertion schemes

A parallel embedding overlapped iterative (EOI) algorithm about classicimplicit equations with asymmetric Saul'yev schemes (CIS-EOI) to solve one-dimensional diffusionequations is discussed to improve the properties of the segment classic implicit iterative (SCII)algorithm. The structure of CIS-EOI method is given and the stability of scheme and convergence ofiteration are proved by matrix method. The property of gradual-approach convergence is alsodiscussed. It has been shown that the convergent rate is faster and the property of gradual-approachconvergence also becomes better with the increasing of the net point in subsystems than with theSCII algorithm. The simulation examples show that the parallel iterative algorithm with a differentinsertion scheme CIS-EOI is more effective.

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)).

THE UNIFORMLY CONVERGENT DIFFERENCE SCHEMES FOR A SINGULAR PERTURBATION PROBLEM OF A SELFADJOINT ORDINARY DIFFERENTIAL EQUATION

In this paper, we construct a class of difference schemes with fitted factors for a singular perturbation problem of a self-adjoint ordinary differential equation. Using a different method from [1], by analyzing the truncation errors of schemes, we give the sufficient conditions under which the solution of lite difference scheme converges uniformly to the solution of the differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence, and applying them to example, obtain the numerical results.

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.

Analysis of the Effect of Different Treatment Schemes on Helicobacter Pylori Infection on Improving the Quality of Life of Patients

Objective:To analyze the effect of different treatment schemes on the quality of life of patients with Helicobacter pylori infection.Methods:From June 2019 to November 2020,96 patients with Helicobacter pylori infection were selected and randomly assigned Combined treatment with omeprazole,amoxicillin and clarithromycin was named group A;Group B was treated with omeprazole,amoxicillin,clarithromycin and colloidal bismuth pectin;Group C was given omeprazole,amoxicillin,clarithromycin and metronidazole.Results:The symptom score and gastroscope score of group B or C were lower than those of group A(P<0.05);The incidence of adverse reactions and HP eradication rate in group C were lower than those in group B,and those in group B were lower than those in group A(P<0.05);The quality of life score of group C was higher than that of group B,and that of group B was higher than that of group A,P<0.05.Conclusions:The sequential therapy of omeprazole,amoxicillin,clarithromycin and metronidazole in patients with Helicobacter pylori infection can improve the HP eradication rate,improve the quality of life of patients,which is safe and efficient.

An Explicit Difference Scheme with High Accuracy and Branching Stability for Solving Parabolic Partial Differential Equation

This paper presents an explicit difference scheme with accuracy and branching stability for solving onedimensional parabolic type equation by the method of undetermined parameters and its truncation error is O(△t4+△x4). The stability condition is r=a△t/△x2<1/2.

HIGH RESOLUTION POSITIVITY-PRESERVING DIFFERENCE SCHEMES FOR TWO DIMENSIONAL EULER EQUATIONS

A class of high resolution positivity preserving Boltzmann type difference schemes for one and two dimensional Euler equations is studied. First, the relation between Boltzmann and Euler equations is analyzed. By using a kind of special interpolation, the high resolution Boltzmann type difference scheme is constructed. Finally, numerical tests show that the schemes are effective and useful.

Stability of Difference Schemes with Intrinsic Parallelism for Quasilinear Parabolic Systems

In this paper, the first boundary problem of quasilinear parabolic system of second order is studied by the finite difference method with intrinsic parallelism. for the problem, the stability of the difference schemes with intrinsic parallelism are justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete data of the original problem, without assuming the existence of the smooth solutions for the origillal problem.

A CONSERVATIVE DIFFERENCE SCHEME FOR CONSERVATIVE DIFFERENTIAL EQUATION WITH PERIODIC BOUNDARY

The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, it is proved that the scheme converges uniformly to the solution of differential equation with order one.

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.

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.

A CLASS OF COMPACT UPWIND TVD DIFFERENCE SCHEMES

A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.

Computational Stability of the Explicit Difference Schemes of the Forced Dissipative Nonlinear Evolution Equations

The computational stability of the explicit difference schemes of the forced dissipative nonlinear evolution equations is analyzed and the computational quasi-stability criterion of explicit difference schemes of the forced dissipative nonlinear atmospheric equations is obtained on account of the concept of computational quasi-stability, Therefore, it provides the new train of thought and theoretical basis for designing computational stable difference scheme of the forced dissipative nonlinear atmospheric equations. Key words Computational quasi-stability - Computational stability - Forced dissipative nonlinear evolution equation - Explicit difference scheme This work was supported by the National Outstanding Youth Scientist Foundation of China (Grant No. 49825109), the Key Innovation Project of Chinese Academy of Sciences (KZCX1-10-07), the National Natural Science Foundation of China (Grant Nos, 49905007 and 49975020) and the Outstanding State Key Laboratory Project (Grant No. 40023001).

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.

Construction of Explicit Quasi-complete Square Conservative Difference Schemes of Forced Dissipative Nonlinear Evolution Equations

Based on the forced dissipetive nonlinear evolution equations for describing the motion of atmosphere and ocean, the computational stability of the explicit difference schemes of the forced dissipotive nonlinear atmospheric and oceanic equations is analyzed and the computationally stable explicit complete square conservative difference schemes are constructed. The theoretical analysis and numerical experiment prove that the explicit complete square conservative difference schemes are computationally stable and deserve to be disseminated.

A Class of High Accuracy Explicit Difference Schemes for Solving the Heat-conduction Equation of High-dimension

In this paper, a class of explicit difference schemes with parameters for solving five-dimensional heat-conduction equation are constructed and studied.the truncation error reaches O（τ^2＋ h%4）, and the stability condition is given. Finally, the numerical examples and numerical results are presented to show the advantage of the schemes and the correctness of theoretical analysis.

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.

Properties of High-Order Finite Difference Schemes and Idealized Numerical Testing

Construction of high-order difference schemes based on Taylor series expansion has long been a hot topic in computational mathematics, while its application in comprehensive weather models is still very rare. Here, the properties of high-order finite difference schemes are studied based on idealized numerical testing, for the purpose of their application in the Global/Regional Assimilation and Prediction System(GRAPES) model. It is found that the pros and cons due to grid staggering choices diminish with higher-order schemes based on linearized analysis of the one-dimensional gravity wave equation. The improvement of higher-order difference schemes is still obvious for the mesh with smooth varied grid distance. The results of discontinuous square wave testing also exhibits the superiority of high-order schemes. For a model grid with severe non-uniformity and non-orthogonality, the advantage of high-order difference schemes is inapparent, as shown by the results of two-dimensional idealized advection tests under a terrain-following coordinate. In addition, the increase in computational expense caused by high-order schemes can be avoided by the precondition technique used in the GRAPES model. In general, a high-order finite difference scheme is a preferable choice for the tropical regional GRAPES model with a quasi-uniform and quasi-orthogonal grid mesh.

DIFFERENCE SCHEME FOR AN INITIAL-BOUNDARY VALUE PROBLEM FOR LINEAR COEFFICIENT-VARIED PARABOLIC DIFFERENTIAL EQUATION WITH A NONSMOOTH BOUNDARY LAYER FUNCTION

In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.