共查询到20条相似文献,搜索用时 15 毫秒
1.
P. N. Vabishchevich 《Computational Mathematics and Mathematical Physics》2010,50(1):112-123
In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are
obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to
time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods
are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the
finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements,
extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues
concerning the computational implementation of schemes having higher approximation orders are discussed. From the general
point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed. 相似文献
2.
The paper investigates some nonconforming finite elements and nonconforming finite element schemes for solving an advection—diffusion equation. This investigation is aimed at finding new schemes for solving parabolic equations. The study uses a finite element method, variational-difference schemes, and test calculations. Two types of schemes are examined: one is obtained with the help of the Bubnov—Galerkin method from a weak problem determination (nonmonotone scheme), and the other one is a monotone up-stream scheme obtained from an approximate weak problem determination with a special approximation of the skew-symmetric terms. 相似文献
3.
Boundary value problems for time-dependent convection-diffusion-reaction equations are basic models of problems in continuum mechanics. To study these problems, various numerical methods are used. With a finite difference, finite element, or finite volume approximation in space, we arrive at a Cauchy problem for systems of ordinary differential equations whose operator is asymmetric and indefinite. Explicit-implicit approximations in time are conventionally used to construct splitting schemes in terms of physical processes with separation of convection, diffusion, and reaction processes. In this paper, unconditionally stable schemes for unsteady convection-diffusion-reaction equations are constructed with explicit-implicit approximations used in splitting the operator reaction. The schemes are illustrated by a model 2D problem in a rectangle. 相似文献
4.
Abumoslem Mohammadi 《Journal of Difference Equations and Applications》2013,19(7):719-728
In this study, first, three non-standard implicit finite difference schemes are proposed for solving the initial-boundary value problem involving a quartic non-linearity that arises in heat transfer involving conduction with thermal radiation. A thin finite rod exposed to radiating heat across its lateral surface into a medium of constant temperature and convection is ignored. Stability and consistency of the third scheme is proved. Numerical results are compared with non-standard explicit finite difference schemes that show fully stability of our third proposed scheme. Then, three non-standard implicit and three non-standard explicit finite difference schemes are proposed for solving the heat transfer problem with additional convection term. It is shown that in the second case when the model involves conduction, radiation and convection terms, the rod reaches steady state sooner. Numerical results for implicit and explicit schemes are compared and the effect of the convection term is discussed. 相似文献
5.
6.
Dan-ping Yang 《计算数学(英文版)》2002,20(2):153-164
1. IntroductionA large number of physical phenomena are modeled by partial differelltial equations orsystems of parabolic type in an evolutionary or eIliptic type at steady state. It is frequentlythe case that a good approximation of some function of the … 相似文献
7.
Xinlong Zhou 《Proceedings of the American Mathematical Society》2006,134(3):859-869
We study the convergence of multivariate subdivision schemes with nonnegative finite masks. Consequently, the convergence problem for the multivariate subdivision schemes with nonnegative finite masks supported on centered zonotopes is solved. Roughly speaking, the subdivision schemes defined by these masks are always convergent, which gives an answer to a question raised by Cavaretta, Dahmen and Micchelli in 1991.
8.
Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes. 相似文献
9.
ZHOU Yulin YUAN Guangwei SHEN Longjun Laboratory of Computational Physics Center of Nonlinear Studies Institute of Applied Physics Computational Mathematics Beijing China 《中国科学A辑(英文版)》2004,47(3):453-472
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 相似文献
10.
S. L. Yang 《计算数学(英文版)》1995,13(3):243-249
In this paper, first, modified upwind finite element schemes are presented for two-point value problem, and then a class of modified upwind Taylor finite element schemes are derived for one dimensional linear hyperbolic equation. The main point of the paper is how to consider the upwind property to construct base functions to make the schemes obtained be MmB (or TVD). Numerical experiments are given to show that the method is efficient to solve the discontinuous solutions. 相似文献
11.
Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems 总被引:6,自引:0,他引:6
Dan-Ping Yang. 《Mathematics of Computation》2000,69(231):929-963
Some least-squares mixed finite element methods for convection-diffusion problems, steady or nonstationary, are formulated, and convergence of these schemes is analyzed. The main results are that a new optimal a priori error estimate of a least-squares mixed finite element method for a steady convection-diffusion problem is developed and that four fully-discrete least-squares mixed finite element schemes for an initial-boundary value problem of a nonlinear nonstationary convection-diffusion equation are formulated. Also, some systematic theories on convergence of these schemes are established.
12.
In this paper, some two-grid finite element schemes are constructed for solving the nonlinear Schrödinger equation. With these schemes, the solution of the original problem is reduced to the solution of the same problem on a much coarser grid together with the solutions of two linear problems on a fine grid. We have shown, both theoretically and numerically, that our schemes are efficient and achieve asymptotically optimal accuracy. 相似文献
13.
Mehdi Dehghan 《Journal of Computational Analysis and Applications》2001,3(4):383-398
Three different implicit finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the second-order (5,1) Backward Time Centered Space (BTCS) implicit formula, and the second-order (5,5) Crank-Nicolson implicit finite difference formula and the fourth-order (9,9) implicit scheme. These finite difference schemes are unconditionally stable. The (9,9) implicit formula takes a huge amount of CPU time, but its fourth-order accuracy is significant. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) times needed for each of the methods are discussed and compared. The implicit finite difference schemes use more central processor times than the explicit finite difference techniques, but they are stable for every diffusion number. 相似文献
14.
多孔介质中可压缩可混溶驱动问题的有限体积元法 总被引:2,自引:0,他引:2
马克颖 《高校应用数学学报(A辑)》2005,20(2):161-169
有界区域上多孔介质中可压缩可混溶驱动问题由两个非线性抛物型方程耦合而成:压力方程和饱和度方程均是抛物型方程.运用有限体积元法对两个方程进行数值分析,给出了全离散有限体积元格式,并通过详细的理论分析,得到了近似解与原问题真解的最优H^1模误差估计。 相似文献
15.
《Numerical Methods for Partial Differential Equations》2018,34(6):2301-2315
The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for this problem. They are based on a modification of standard discretizations of time derivatives and, in some cases, allow to obtain the exact solution of problems. For multidimensional problems, we can consider the problem of increasing the accuracy only for the most important components of the approximate solution. In the present work, new unconditionally stable schemes for parabolic problems are constructed, which are exact for the fundamental mode. Such two‐level schemes are designed via a modification of standard schemes with weights using Padé approximations. Numerical results obtained for a model problem demonstrate advantages of the proposed fundamental mode exact schemes. 相似文献
16.
Ujjwal Koley Siddhartha Mishra Nils Henrik Risebro Magnus Svärd 《BIT Numerical Mathematics》2009,49(2):375-395
We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with
the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The
finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation
Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the
stability as well as high order of accuracy of the schemes.
相似文献
17.
1.IntroductionIn[1]and[2]thegeneralfinitedifferenceschemeshavingtheintrinsiccharacterofparallelismfortheboundaryvalueproblemsofthenonlinearparabolicsystemofgeneralform(i.e.,non-divergencetype)arediscussedundertheassumptionthatthereisanuniquesmoothsol... 相似文献
18.
对差分法时程积分的反思 总被引:10,自引:1,他引:9
以往偏微分方程时间步的数值积分主要由有限差分法来执行,然而当时间步长较大时会引起数值不稳定性。本文给出的单点精细积分法导出的显式积分格式可证明是无条件稳定的。就扩散方程与对流─扩散方程作出了本文方法与差分法导出的格式之间的对比。数值例题也表明了单点积分法的优越性。 相似文献
19.
Yulin Zhou 《计算数学(英文版)》1997,15(4):289-310
1.lnthestudyoftheprobleminphysics,mechanics,chemicalreactions,biologyandotherpracticalsciences,thelinearandnonlinearparabolicequationsandsystemsareappearedveryfrequently.Manynumericalinvestigationsinscientificandengineeringproblemsespeciallyinthelargescalecomputationalproblemsoftencontainthenumer-icalsolutionsofparabolicequationsandsystems.ThemethodwithunequalmeshstePSisnotavoidableinthesecomputations.Manyunexpectedandselfcontradictoryphe-nomenonraisingfromtheuseofunequalmeshstepscallourgreata… 相似文献
20.
《Applied Mathematical Modelling》2001,25(9):743-754
An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-difference schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centred space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank–Nicolson implicit method. The classical FTCS explicit formula and the 5-point FTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank–Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference mehods. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed. 相似文献