Four-point explicit difference schemes for the dispersive equation

A class of three-level explicit difference schemes for the dispersive equationu_1=au_(xxx)are established These schemes have higher stability and involve four meshpoints at the middle level.Their local truncation errors are O(τ+h)and stabilityconditions are from|R|≤0.25 to|R|≤10,where|R|=|a|τ/h~3,which is muchbetter than|R|≤0.25.     

The high accuracy explicit difference scheme for solving parabolic equations 3-dimension

ＩｎｔｒｏｄｕｃｔｉｏｎＦｒｏｍｐｒａｃｔｉｃａｌｐｒｏｂｌｅｍ,ｗｅｃｏｕｌｄｃｏｎｃｌｕｄｅｍａｎｙｐｒｏｂｌｅｍｓａｂｏｕｔｓｏｌｖｉｎｇｐａｒａｂｏｌｉｃｐａｒｔｉａｌｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ．Ｎｏｗｔｈｅｒｅａｒｅｍａｎｙｎｕ...     

The stability of difference schemes of a higher dimensional parabolic equation

This paper proposes a new method to improve the stability condition of differencescheme of a parabolic equation.Necessary and sufficient conditions of the stability of thisnew method are given and proved.Some numerical examples show that this method hassome calculation advantages.     

一族新的高精度显式差分格式

对求解三维势物型偏微分方程,利用待定参数法构造出一族新的高精度的三层显式差分格式,其精度为Ｏ（△ｔ＾３＋△ｘ＾４＋△ｙ＾４＋△ｚ＾４）,并论证了其稳定性,通过数值实例可见其精度较文「１」提高２位有效数字。     

色散方程的高稳定性两层四点显格式的单点精细积分法

基于单点精细积分的思想,对色散方程Ｕｔ＝ａＵｘｘｘ构造了一类高稳定性的两层四点显式差分格式,其局部截断误差为Ｏ（τ＋ｈ）稳定性条件为│Ｒ│＝│ａτ／ｈ＾３│≤ｆ（β）,对任意正实数β为单调递增函数,它们不仅显著地改善了同类格式的稳定性条件│Ｒ│≤０．２５而且也优于众多三层多点（５点或５点以上）显格式的稳定性条件。     

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

Ａ－ＨＩＧＨ－ＯＲＤＥＲＡＣＣＵＲＡＣＹＥＸＰＬＩＣＩＴＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＳＯＬＶＩＮＧＴＨＥＥＱＵＡＴＩＯＮＯＦＴＷＯ－ＤＩＭＥＮＳＩＯＮＡＬＰＡＲＡＢＯＬＩＣＴＹＰＥＭａＭｉｎｇｓｈｕ（马明书）（ＲｅｃｅｉｖｅｄＪｕｎｅ２,１...     

色散方程的高稳定性三层五点显格式的广义单点精细积分法

基于文（１）中的单点精细积分方法,对色散方程Ｕｔ＝ａＵｘｘｘ提出了一种构造高稳定性三层五点（蛙跳）显格式的广义单点精细积分法,文中格式的局部截断误差为Ｏ（ｘ＾２＋ｈ＾２）,而稳定性条件为｜Ｒ｜≤ｇ（β）（其中ｇ对任意正实数是单调递增函数）,同时类格式中最好的。     

A family of high-order accuracy explicit difference schemes with branching stability for solving 3-D parabolic partial differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｏｆｔｅｎｍｅｅｔｔｈｅｐｒｏｂｌｅｍｏｆｓｏｌｖｉｎｇｅｑｕａｔｉｏｎｏｆｐａｒａｂｏｌｉｃｔｙｐｅｉｎｍａｎｙｆｉｅｌｄｓｓｕｃｈａｓｓｅｅｐａｇｅ ,ｄｉｆｆｕｓｉｏｎ ,ｈｅａｔｃｏｎｄｕｃｔｉｏｎａｎｄｓｏｏｎ .Ｉｎｔｈｅｃａｓｅｏｆ3＿ｄｉｍｅｎｓｉｏｎ ,ｔｈｅｍｏｄｅｌｉｓａｎｉｎｉｔｉａｌａｎｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍａｓｆｏｌｌｏｗｓ: ｕ ｔ = 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 　　　　 (0 <ｘ,ｙ,ｚ<1 ;ｔ>0 ) ,ｕ(ｘ ,ｙ,ｚ,0 ) =φ(ｘ ,ｙ ,ｚ)…     

The 3-layered explicit difference scheme for 2-D heat equation

A 3-layered explicit difference scheme for the numerical solution of 2-D heat equation is proposed. Firstly, a general symmetric difference scheme is constructed and its optimal error is obtained. Then two kinds of condition for choosing the parameters for optimal error and stable difference scheme are given. Finally, some numerical results are presented to show the advantage of the schemes Foundation items: the Science Foundation of Chinese Postdoctoral (2002031224); the Science Foundations of Southeast University (9209011148, 3007011043) Biography: Liu Ji-jun (1965-)     

A new high-order accuracy explict difference scheme for solving three-dimensional parabolic equations

In this paper, a new three-level explicit difference scheme with high-orderaccuracy is proposed for solving three-dimensional parabolic equations. The stabilitycondition is r＝△t/△x² ＝△t/△y²＝△t/△z²≤1/4, and the trumcation error is O(△t²+△x⁴).     

A compact finite difference method for solving Burgers' equation

In this paper, a high‐order accurate compact finite difference method using the Hopf–Cole transformation is introduced for solving one‐dimensional Burgers' equation numerically. The stability and convergence analyses for the proposed method are given, and this method is shown to be unconditionally stable. To demonstrate efficiency, numerical results obtained by the proposed scheme are compared with the exact solutions and the results obtained by some other methods. The proposed method is second‐ and fourth‐order accurate in time and space, respectively. Copyright © 2009 John Wiley & Sons, Ltd.     

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. Partly supported by the State Major Key Project for Basic Researches of China Worked as a post doctor in Computing Center, Chinese Academy of Sciences, when this paper was submitted     

Criteria for finite element algorithm of generalized heat conduction equation

To eliminate oscillation and overbounding of finite element solutions of classical heat conduction equation, the author and Xiao have put forward two new concepts of monotonies and have derived and proved several criteria. This idea is borrowed here to deal with generalized heat conduction equation and finite element criteria for eliminating oscillation and overbounding are also presented. Some new and useful conclusions are drawn.     

Dissipation matrix and artificial heat conduction for Godunov‐type schemes of compressible fluid flows

This paper aims to reassess the Riemann solver for compressible fluid flows in Lagrangian frame from the viewpoint of modified equation approach and provides a theoretical insight into dissipation mechanism. It is observed that numerical dissipation vanishes uniformly for the Godunov‐type schemes in the sense that associated dissipation matrix has zero determinant if an exact or approximate Riemann solver is used to construct numerical fluxes in the Lagrangian frame. This fact connects to some numerical defects such as the wall‐heating phenomenon and start‐up errors. To cure these numerical defects, a traditional numerical viscosity is added, as well as the artificial heat conduction is introduced via a simple passage of the Lax–Friedrichs type discretization of internal energy. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

A novel optimization technique for explicit finite‐difference schemes with application to AeroAcoustics

The present paper addresses the optimization of finite‐difference schemes when these are to be used for numerically approximating spatial derivatives in aeroacoustics evolution problems. With that view in mind, finite‐difference operators are firstly detailed from a theoretical point of view. Secondly, time, the way such operators can be optimized in a spectral‐like sense is recalled, before the main limitations of such an optimization are highlighted. This leads us to propose an alternative optimization approach of innovative character. Such a novel optimization technique consists of enhancing the scheme's formal accuracy through a minimization of its leading‐order truncation error. This so‐called intrinsic optimization procedure is first detailed, before it is thoroughly analyzed, from both a theoretical and a practical point of view. The second part of the paper focuses on two particular intrinsically optimized schemes, which are carefully assessed via a direct comparison against their standard and/or spectral‐like optimized counterparts, such a comparative exercise being conducted utilizing several academic test cases of increasing complexity. There, it is shown how intrinsically optimized schemes indeed constitute an advantageous alternative to either the standard or the spectral‐like optimized ones, being allotted with both (i) the better scalability of the former scheme with respect to grid convergence effects when the grid density increases and (ii) the higher accuracy of the latter scheme when the discretization level becomes marginal. Thanks to that, such intrinsically optimized schemes offer very good trade‐offs in terms of (i) accuracy; (ii) robustness; and (iii) numerical efficiency (CPU cost). Copyright © 2015 John Wiley & Sons, Ltd.     

The similar solutions of nonlinear heat conduction equation

ＴＨＥＳＩＭＩＬＡＲＳＯＬＵＴＩＯＮＳＯＦＮＯＮＬＩＮＥＡＲＨＥＡＴＣＯＮＤＵＣＴＩＯＮＥＱＵＡＴＩＯＮＹｕａｎＹｉｗｕ（袁镒吾）（ＣｅｎｔｒａｌＳｏｕｔｈＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ,Ｃｈａｎｇｓｈａ４１００１２．Ｐ．Ｒ．Ｃｈｉｎａ...     

求解二维污染扩散方程的时空任意阶的三点紧致显格式

通过在泰勒级数展开中运用逐阶迭代的方法,推导出了空间二阶导数任意精度的三点紧致的表达式,并在半高散方程中通过二维扩散方程本身把时间导数转换为空间导数,从而推导出了时空任意阶的三点紧致显格式.数值实验表明,本文格式的精度很高,而且具有使用简单,易于编程的优点,对求解二维污染扩散方程具有很好的应用前景.     

A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations

In this paper, a fully third-order accurate projection method for solving the incompressible Navier-Stokes equations is proposed. To construct the scheme, a continuous projection procedure is firstly presented. We then derive a sufficient condition for the continuous projection equations to be temporally third-order accurate approximations of the original Navier-Stokes equations by means of the local- truncation-error-analysis technique. The continuous projection equations are discretized temporally and spatially to third-order accuracy on the staggered grids, resulting in a fully third-order discrete projection scheme. The possibility to design higher-order projection methods is thus demonstrated in the present paper. A heuristic stability analysis is performed on this projection method showing the probability of its being stable. The stability of the present scheme is further verified through numerical tests. The third-order accuracy of the present projection method is validated by several numerical test cases. The project supported by the China NKBRSF (2001CB409604) The English text was polished by Yunming Chen