UNCONDITIONAL STABLE DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR SEMILINEAR PARABOLIC SYSTEMS OF DIVERGENCE TYPE

The general finite difference schemes with intrinsic parallelism for the boundary value problem of the semilinear parabolic system of divergence type with bounded measurable coefficients is studied. By the approach of the discrete functional analysis, the existence and uniqueness of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the unconditional stability of the general difference schemes with intrinsic parallelism justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete initial data of the original problems in the discrete W_2~(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the certain difference schemes with intrinsic parallelism to the unique generalized solution of the original semilinear parabolic problem is proved.     

The unconditional stable and convergent difference methods with intrinsic parallelism for quasilinear parabolic systems

A kind of the general finite difference schemes with intrinsic parallelism forthe boundary value problem of the quasilinear parabolic system is studied without assum-ing heuristically that the original boundary value problem has the unique smooth vectorsolution. By the method of a priori estimation of the discrete solutions of the nonlineardifference systems, and the interpolation formulas of the various norms of the discretefunctions and the fixed-point technique in finite dimensional Euclidean space, the exis-tence and uniqueness of the discrete vector solutions of the nonlinear difference systemwith intrinsic parallelism are proved. Moreover the unconditional stability of the generalfinite difference schemes with intrinsic parallelism is justified in the sense of the continu-ous dependence of the discrete vector solution of the difference schemes on the discretedata of the original problems in the discrete w_2~(2,1) norms. Finally the convergence of thediscrete vector solutions of the certain differe     

General difference schemes with intrinsic parallelism for nonlinear parabolic systems

The boundary value problem for nonlinear parabolic system is solved by the finite difference method with intrinsic parallelism. The existence of the discrete vector solution for the general finite difference schemes with intrinsic parallelism is proved by the fixed-point technique in finite-dimensional Euclidean space. The convergence and stability theorems of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. The limitation vector function is just the unique generalized solution of the original problem for the parabolic system.     

THE UNCONDITIONAL CONVERGENT DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR QUASILINEAR PARABOLIC SYSTEMS WITH TWO DIMENSIONS

In the present work we are going to solve the boundary value problem for the quasilinear parabolic systems of partial differential equations with two space dimensions by the finite difference method with intrinsic parallelism.Some fundamental behaviors of general finite difference schemes with intrinsic parallelism for the mentioned problems are studied.By the method of a priori estimation of the discrete solutions of the nonlinear difference systems,and the interpolation formulas of the various norms of the discrete functions and the fixed-point technique in finite dimensional Euclidean space,the existennce of the discrete vector solutions of the nonliear difference system with intrinsic parallelism are proved .Moreover the convergence of the discrete vector solutions of these difference schemes to the unique generalizd solution of the original quasilinear parabolic problem is proved.     

Unconditional stability of parallel alternating difference schemes for semilinear parabolic systems

The general alternating schemes with intrinsic parallelism for semilinear parabolic systems are studied. First we prove the a priori estimates in the discrete H¹ space of the difference solution for these schemes. Then the existence of the difference solution for these schemes follows from the fixed point principle. Finally the unconditional stability of the general alternating schemes is proved. The alternating group explicit scheme, the alternating segment explicit–implicit scheme and the alternating segment Crank–Nicolson scheme are the special cases of the general alternating schemes.     

Parallel difference schemes with interface extrapolation terms for quasi-linear parabolic systems

In this paper some new parallel difference schemes with interface extrapolation terms for a quasi-linear parabolic system of equations are constructed. Two types of time extrapolations are proposed to give the interface values on the interface of sub-domains or the values adjacent to the interface points, so that the unconditional stable parallel schemes with the second accuracy are formed. Without assuming heuristically that the original boundary value problem has the unique smooth vector solution, the existence and uniqueness of the discrete vector solutions of the parallel difference schemes constructed are proved. Moreover the unconditional stability of the parallel difference schemes is justified in the sense of the continuous dependence of the discrete vector solution of the schemes on the discrete known data of the original problems in the discrete W2(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the parallel difference schemes with interface extrapolation terms to the unique generalized solution of the original quasi-linear parabolic problem is proved. Numerical results are presented to show the good performance of the parallel schemes, including the unconditional stability, the second accuracy and the high parallelism.     

Difference schemes with intrinsic parallelism for quasi-linear parabolic systems

The boundary value problem for quasi-linear parabolic system is solved by the finite difference method with intrinsic parallelism. The existence and uniqueness and convergence theorems of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. The limiting vector function is just the unique generalized solution of the original problem for the parabolic system. Project supported by the National Natural Science Foundation of China and the Foundation of Chinese Academy of Engineering Physics.     

Difference Method of General Schemes with Intrinsic Parallelism for One-dimensional Quasilinear Parabolic Systems with Bounded Measurable Coefficients

The difference method of the general finite difference schemes with intrinsic parallelism for the boundary value problem of the quasilinear parabolic system is studied without assuming heuristically that the original boundary value problem has the unique smooth vector solution.     

STABILITY OF DIFFERENCE SCHEMES FOR NONLINEAR PARABOLIC SYSTEMS WITH THREE DIMENSIONS

1.Introduction1).Inthepresentworkwearegoingtostudythedeferenceschemesfortheboundaryvalueproblemofthenonlinearparabolicsystemsofpartialdifferentialequationswithtwoandthreespacedimensions.Inthecaseofthreespacedimension,thenonlinearparabolicsystemisoftheformat~A(x,y,z,t,u,aamactu.)(u.. aam u..) f(x,y,z,t,aliojac,u.),(1)whereu=(altuZI',urn)istheunknownm-dimensionalvectorfunction(m3l),A(x,yiitt,u,PI,pz,ps)isagivenmXmmatrixfunction,f(x,y,itt,utpl,p2,p3)isthegivenmdimensionalvectorfunctionandalso…     

GENERAL DIFFERENCE SCHEMES WITH INTRINSIC PARALLELISM FOR SEMILINEAR PARABOLIC SYSTEMS OFDIVERGENCE TYPE

1.IntroductionIn[1]and[2]thegeneralfinitedifferenceschemeshavingtheintrinsiccharacterofparallelismfortheboundaryvalueproblemsofthenonlinearparabolicsystemofgeneralform(i.e.,non-divergencetype)arediscussedundertheassumptionthatthereisanuniquesmoothsol...     

完全非线性伪抛物组的非均匀网格差分格式

完全非线性伪抛物组的非均匀网格差分格式韩永前，袁光伟，周毓麟（北京应用物理与计算数学研究所）ＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＳＷＩＴＨＮＯＮＵＮＩＦＯＲＭＭＥＳＨＥＳＦＯＲＦＵＬＬＹＮＯＮＬＩＮＥＡＲＰＳＥＵＤＯ－ＰＡＲＡＢＯＬＩＣＳＹＳＴＥＭＳ￥Ｈ...     

带扩散项四阶抛物方程的一类本性并行差分格式

给出逼近带扩散项四阶抛物方程一组非对称差分格式,对此组非对称格式重新组合,得到了一类新的具有并行本性的算法.随后,利用矩阵法证明了算法的绝对稳定性.最后给出数值实验.     

PARALLEL COMPUTING METHOD OF PURE ALTERNATIVE SEGMENT EXPLICIT-IMPLICIT DIFFERENCE SCHEME FOR NONLINEAR LELAND EQUATION

The research on the numerical solution of the nonlinear Leland equation has important theoretical significance and practical value. To solve nonlinear Leland equation, this paper offers a class of difference schemes with parallel nature which are pure alternative segment explicit-implicit(PASE-I) and implicit-explicit(PASI-E) schemes. It also gives the existence and uniqueness,the stability and the error estimate of numerical solutions for the parallel difference schemes. Theoretical analysis demonstrates that PASE-I and PASI-E schemes have obvious parallelism, unconditionally stability and second-order convergence in both space and time. The numerical experiments verify that the calculation accuracy of PASE-I and PASI-E schemes are better than that of the existing alternating segment Crank-Nicolson scheme, alternating segment explicit-implicit and implicit-explicit schemes. The speedup of PASE-I scheme is 9.89, compared to classical Crank-Nicolson scheme. Thus the schemes given by this paper are high efficient and practical for solving the nonlinear Leland equation.     

Burgers方程的一类交替分组方法

对于Burgers方程给出了一组新的Saul'yev型非对称差分格式，并用这些差分格式构造了求解非线性Burgers方程的交替分组四点方法．该算法把剖分节点分成若干组，在每组上构造能够独立求解的差分方程．因此算法具有并行本性，能直接在并行计算机上使用．章还证明了所给算法线性绝对稳定．数值试验表明，该方法使用简便，稳定性好，有很好的精度。     

色散方程的一类本性并行的差分格式

对一维色散方程给出了本性并行的一般的交替差分格式，证明了该类格式的绝对稳定性已有的交替分组显格式(AGE)是该类格式的特例．作为特例，进一步得到交替分段显一隐格式(ASF-I)和交替分段Crank-Nicolson格式(ASC-N)．数值实验比较了这几个格式数值解的精确性．     

Difference Schemes with Nonuniform Meshes for Nonlinear Parabolic System

1.Introduction1.Fromtheverybeginningofsixtiestothelateeighties,therearemanywerkscontributedtothestudiesoftheboundaryproblemsandinitialvalueproblemsfortheordinarydifferentialequationsbythemethodofdifferenceschemeswithnonuniform.eshesl1-4l.Butitisextremelyrareontheworksconcerningtotheanalysisoffinitedifferenceschemeswithnonuniformmeshesfortheproblemsofpartialdifferentialequations.Byusingofthedifferenceschemeswithnonuniformmeshesapprotimationfortheproblemsofpartialdifferentialequationsthereareman…     

An Explicit-Implicit Predictor-Corrector Domain Decomposition Method for Time Dependent Multi-Dimensional Convection Diffusion Equations

The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms. In this work, we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique, modified upwind differences with explicitimplicit coupling, the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy. Moreover, for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictorcorrector or stabilized schemes. These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments.     

Finite Difference Method with Nonuniform Meshes for Quasilinear Parabolic Systems

1.lnthestudyoftheprobleminphysics,mechanics,chemicalreactions,biologyandotherpracticalsciences,thelinearandnonlinearparabolicequationsandsystemsareappearedveryfrequently.Manynumericalinvestigationsinscientificandengineeringproblemsespeciallyinthelargescalecomputationalproblemsoftencontainthenumer-icalsolutionsofparabolicequationsandsystems.ThemethodwithunequalmeshstePSisnotavoidableinthesecomputations.Manyunexpectedandselfcontradictoryphe-nomenonraisingfromtheuseofunequalmeshstepscallourgreata…     

三维两相渗流驱动问题迎风区域分裂显隐差分法

对三维两相渗流驱动问题提出了两种迎风区域分裂显隐差分格式．压力方程采用了七点差分格式,为了能达到实际并行计算的要求,对饱和度方程采用了迎风区域分裂差分法,内边界处和各子区域分别对应显隐格式．得到了离散l2模收敛性分析,最后给出数值试验,支撑了理论分析结果．     

Unconditional stability of alternating difference schemes with intrinsic parallelism for two‐dimensional parabolic systems

The difference method with intrinsic parallelism for two dimensional parabolic system is studied. The general alternating difference schemes, in particular those with variable time steplengthes, are constructed and proved to be unconditionally stable. The two dimensional alternating group explicit scheme, alternating block explicit‐implicit scheme, alternating block Crank‐Nicolson scheme and block ADI scheme are the special cases of the general schemes constructed here. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 625–636, 1999