高精度交替方向隐式差分法的理论与应用
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摘要
自然科学、工程技术、社会科学中存在着大量的偏微分方程(PDEs).然而，许多PDEs的真解很难得到，或以实用的表达式表出.因此，为获得PDEs的近似解，发展高性能的PDEs数值解法是十分必要的.交替方向隐式方法(ADI)将高维问题的求解转换成一系列一维问题的求解，提高了计算效率.高阶紧格式(HOC)是一类用较少的计算节点就可达到高精度的差分方法.高阶紧交替方向隐式(HOC ADI)法是这两类方法的结合，综合了它们的优点.多年来，这类方法一直受到人们的普遍关注.本文以一些PDEs为例研究几类HOC ADI法的理论性质.
第一章介绍了数值方法的历史与现状，给出了问题的来源，本文的主要工作，与时空离散相关的记号和一些引理.
第二章给出了二维线性双曲方程条件稳定的HOC ADI格式.运用von Neumann法获得了此算法的稳定条件.在满足稳定条件下，运用能量方法证明了它在H~1-和L~2-范数下有O(△t~4+h_x~4+h_y~4)阶精度.尽管该算法是条件稳定的，但是稳定条件还好.只要根据稳定条件适当地选取网格步长，数值解能以最优的收敛速度快速收敛到真解.另外，我们类似地发展了三维情形的紧ADI格式，并给出了一些理论结果.
第三章研究了非线性波动方程的三层HOC ADI格式.在一四阶格式逼近第一层真解的情况下，运用能量法获得了HOC ADI法在不同范数下的收敛阶.建立了对应的外推算法以提高计算效率，并通过引入两个辅助问题，给出了外推解的收敛性.
第四章讨论了阻尼波动方程的一族三层HOC ADI方法，及其相应的外推算法.运用与前一章类似的分析方法，获得了数值解及其外推解在不同范数下的收敛阶.
第五章研究了线性抛物方程的高阶紧多步分裂(HOC MFS)方法.将给出的一些引理与能量法结合可证其解在L~2-, H~1-和L~∞-范数下以O(△t~2+h_x~4+h_y~4)的收敛率无条件收敛.此外，建立一类外推算法，得到了关于L~2-, H~1-和L~∞-范数有O(△t~4+h_x~4+h_y~4)阶精度的外推解.最后，通过引入一新变换，将这些算法推广到了常系数对流扩散方程.
第六章提出了二维非线性粘弹性波方程的三层HOC MFS方法.首先，引入两个变量，将黏弹性波动方程转化成与抛物方程等价的形式.然后，对等价形式构建HOC MFS方法及其外推算法.此外，算法的理论分析也得到具体地研究.
第七章提出了三维非线性粘弹性波方程的两层Crank-Nicolson HOC ADI方法.这类ADI方法在x-和y-方向上只需解系数矩阵为三对角阵的线性方程组，在z-方向上需解非线性方程组.运用能量法，这类HOC ADI方法在H~1-范数下有O(△t~2+h_x~4+h_y~4)阶精度.为了改进时间精度，我们提出了基于两个时间网格的Richardson外推算法.通过引入一辅助问题，我们严格证明了外推解的收敛性.
第八章归纳了本文的主要贡献，结论和展望.
数值结果验证了文中算法的性能，也表明了相关理论的正确性.另外，文中的算法可以推广于其它PDEs的求解.
Partial differential equations (PDEs) are frequently encountered in science, engineerand society. However, it is difficult to get exact solutions, or to give practical expressions ofexact solutions. Hence, for obtaining good approximate solutions, it is necessary to devel-op high performance numerical algorithms for PDEs. Alternating direction implicit (ADI)methods can reduce the solution of a multidimensional problem to series of independentone-dimensional problems, and thus save time cost. High-order compact (HOC) schemesis a kind of difference methods, which can use less nodes to attain high accuracy. Combi-nations of ADI methods with HOC schemes yield HOC ADI methods, which preserve theadvantages of HOC schemes and high performance of ADI methods, and thus have attract-ed much attention over the years. Taking some PDEs for example, we study the theoreticalproperties of some HOC ADI methods in this dissertation.
In chapter1, we introduce backgrounds and developments of numerical methods, andgive research motivation, main works, notations associated to spatial and temporal grids,and some lemmas.
In chapter2, a conditionally stable HOC ADI method for a linear hyperbolic equationwith two spatial variables is derived. Its stability criterion is determined by using von Neu-mann method. It is shown through a discrete energy method that this method can attainfourth-order accuracy in both time and space with respect to H~1-and L~∞-norms providedthe stability is fulfilled. Although the method is conditionally stable, stable restriction is notbad. Numerical solutions can fast converge to exact solution only if that spatial and tempo-ral mesh-sizes are suitably chosen according to stability criterion. In addition, extension ofHOC ADI method to3D case is also discussed, and theoretical results are similarly given.
In chapter3, a new three-level HOC ADI method for a nonlinear wave equation isproposed. Basing on a fourth-order approximation to the exact solution at the first timelevel, convergence orders of HOC ADI method in different norms are obtained. A class ofRichardson extrapolation formulas is developed to improve computational efficiency. By in-troducing two auxiliary problems, we give the convergence rates of extrapolation solutions.
Chapter4focuses on the construction and extrapolation of a family of three-level HOC ADI methods for a damped wave equation. Applying the methods similar to those usedin Chapter3, the convergence orders of numerical solution and extrapolation solution indifferent norms are obtained.
Chapter5is primarily aimed at the analysis of a high-order compact multi-step frac-tional steps (HOC MFS) method for a linear parabolic equation. A combination of lemmasproposed with energy method can prove the discrete solution is unconditionally convergentwith an order of O(△t~2+h_x~4+h_y~4) in H~1-, L~2-and L~∞-norms. Moreover, a class of Richard-son extrapolation methods is established to obtain the extrapolation solution of order four inboth time and space with respect to H~1-, L~2-and L~∞-norms. By using a new transforma-tion, HOC MFS method is easily generalized to convention diffusion equation with constantcoefficients.
In Chapter6, we propose a HOC MFS method with three-level for a two-dimensional(2D) nonlinear viscous wave equation. First, we introduce two auxiliary functions to trans-form this equation into an equivalent form similar to parabolic equation. Then, HOC MFSmethod and corresponding extrapolation algorithms for the equivalent form are established.Convergence analysis of the algorithms is carried out in detail.
In Chapter7, a Crank-Nicolson HOC ADI method for a three-dimensional (3D) non-linear viscous wave equation is derived. We need to solve the system of linear tridiagonalalgebra equations in x-and y-directions, and need to solve the system of nonlinear algebraequations in z-direction, respectively. By the discrete energy method, it is shown that thismethod has convergence order of O(△t~2+h_x~4+h_y~4) in H~1-norm. A two-grid ex-trapolation method is developed to improve temporal accuracy. By introducing an auxiliaryproblem, we can prove the convergence rate of extrapolation solution.
Chapter8is devoted to the conclusions of main contributions, main results and someresearches in future.
Numerical results testify the performance of the algorithms and support theoreticalanalysis. In addition, the algorithms proposed in this dissertation can be easily extended toother PDEs.
第一章介绍了数值方法的历史与现状，给出了问题的来源，本文的主要工作，与时空离散相关的记号和一些引理.
第二章给出了二维线性双曲方程条件稳定的HOC ADI格式.运用von Neumann法获得了此算法的稳定条件.在满足稳定条件下，运用能量方法证明了它在H~1-和L~2-范数下有O(△t~4+h_x~4+h_y~4)阶精度.尽管该算法是条件稳定的，但是稳定条件还好.只要根据稳定条件适当地选取网格步长，数值解能以最优的收敛速度快速收敛到真解.另外，我们类似地发展了三维情形的紧ADI格式，并给出了一些理论结果.
第三章研究了非线性波动方程的三层HOC ADI格式.在一四阶格式逼近第一层真解的情况下，运用能量法获得了HOC ADI法在不同范数下的收敛阶.建立了对应的外推算法以提高计算效率，并通过引入两个辅助问题，给出了外推解的收敛性.
第四章讨论了阻尼波动方程的一族三层HOC ADI方法，及其相应的外推算法.运用与前一章类似的分析方法，获得了数值解及其外推解在不同范数下的收敛阶.
第五章研究了线性抛物方程的高阶紧多步分裂(HOC MFS)方法.将给出的一些引理与能量法结合可证其解在L~2-, H~1-和L~∞-范数下以O(△t~2+h_x~4+h_y~4)的收敛率无条件收敛.此外，建立一类外推算法，得到了关于L~2-, H~1-和L~∞-范数有O(△t~4+h_x~4+h_y~4)阶精度的外推解.最后，通过引入一新变换，将这些算法推广到了常系数对流扩散方程.
第六章提出了二维非线性粘弹性波方程的三层HOC MFS方法.首先，引入两个变量，将黏弹性波动方程转化成与抛物方程等价的形式.然后，对等价形式构建HOC MFS方法及其外推算法.此外，算法的理论分析也得到具体地研究.
第七章提出了三维非线性粘弹性波方程的两层Crank-Nicolson HOC ADI方法.这类ADI方法在x-和y-方向上只需解系数矩阵为三对角阵的线性方程组，在z-方向上需解非线性方程组.运用能量法，这类HOC ADI方法在H~1-范数下有O(△t~2+h_x~4+h_y~4)阶精度.为了改进时间精度，我们提出了基于两个时间网格的Richardson外推算法.通过引入一辅助问题，我们严格证明了外推解的收敛性.
第八章归纳了本文的主要贡献，结论和展望.
数值结果验证了文中算法的性能，也表明了相关理论的正确性.另外，文中的算法可以推广于其它PDEs的求解.
Partial differential equations (PDEs) are frequently encountered in science, engineerand society. However, it is difficult to get exact solutions, or to give practical expressions ofexact solutions. Hence, for obtaining good approximate solutions, it is necessary to devel-op high performance numerical algorithms for PDEs. Alternating direction implicit (ADI)methods can reduce the solution of a multidimensional problem to series of independentone-dimensional problems, and thus save time cost. High-order compact (HOC) schemesis a kind of difference methods, which can use less nodes to attain high accuracy. Combi-nations of ADI methods with HOC schemes yield HOC ADI methods, which preserve theadvantages of HOC schemes and high performance of ADI methods, and thus have attract-ed much attention over the years. Taking some PDEs for example, we study the theoreticalproperties of some HOC ADI methods in this dissertation.
In chapter1, we introduce backgrounds and developments of numerical methods, andgive research motivation, main works, notations associated to spatial and temporal grids,and some lemmas.
In chapter2, a conditionally stable HOC ADI method for a linear hyperbolic equationwith two spatial variables is derived. Its stability criterion is determined by using von Neu-mann method. It is shown through a discrete energy method that this method can attainfourth-order accuracy in both time and space with respect to H~1-and L~∞-norms providedthe stability is fulfilled. Although the method is conditionally stable, stable restriction is notbad. Numerical solutions can fast converge to exact solution only if that spatial and tempo-ral mesh-sizes are suitably chosen according to stability criterion. In addition, extension ofHOC ADI method to3D case is also discussed, and theoretical results are similarly given.
In chapter3, a new three-level HOC ADI method for a nonlinear wave equation isproposed. Basing on a fourth-order approximation to the exact solution at the first timelevel, convergence orders of HOC ADI method in different norms are obtained. A class ofRichardson extrapolation formulas is developed to improve computational efficiency. By in-troducing two auxiliary problems, we give the convergence rates of extrapolation solutions.
Chapter4focuses on the construction and extrapolation of a family of three-level HOC ADI methods for a damped wave equation. Applying the methods similar to those usedin Chapter3, the convergence orders of numerical solution and extrapolation solution indifferent norms are obtained.
Chapter5is primarily aimed at the analysis of a high-order compact multi-step frac-tional steps (HOC MFS) method for a linear parabolic equation. A combination of lemmasproposed with energy method can prove the discrete solution is unconditionally convergentwith an order of O(△t~2+h_x~4+h_y~4) in H~1-, L~2-and L~∞-norms. Moreover, a class of Richard-son extrapolation methods is established to obtain the extrapolation solution of order four inboth time and space with respect to H~1-, L~2-and L~∞-norms. By using a new transforma-tion, HOC MFS method is easily generalized to convention diffusion equation with constantcoefficients.
In Chapter6, we propose a HOC MFS method with three-level for a two-dimensional(2D) nonlinear viscous wave equation. First, we introduce two auxiliary functions to trans-form this equation into an equivalent form similar to parabolic equation. Then, HOC MFSmethod and corresponding extrapolation algorithms for the equivalent form are established.Convergence analysis of the algorithms is carried out in detail.
In Chapter7, a Crank-Nicolson HOC ADI method for a three-dimensional (3D) non-linear viscous wave equation is derived. We need to solve the system of linear tridiagonalalgebra equations in x-and y-directions, and need to solve the system of nonlinear algebraequations in z-direction, respectively. By the discrete energy method, it is shown that thismethod has convergence order of O(△t~2+h_x~4+h_y~4) in H~1-norm. A two-grid ex-trapolation method is developed to improve temporal accuracy. By introducing an auxiliaryproblem, we can prove the convergence rate of extrapolation solution.
Chapter8is devoted to the conclusions of main contributions, main results and someresearches in future.
Numerical results testify the performance of the algorithms and support theoreticalanalysis. In addition, the algorithms proposed in this dissertation can be easily extended toother PDEs.
引文
[1] McCalpin J D. A comparison of second-order and fouth-order pressure gradient algorithms in aσ-co-ordinate ocean model. Int. J. Numer. Meth. Fluids,1994,18:361–383.
[2] Adam Y. A hermitian finite difference method for the solution of parabolic equations. Comput.Math. Appl.,1975,1:393–406.
[3] Hirsch R S. High order accurate difference solutions of fluid mechanics problems by a compactdifferencing technique. J. Comput. Phys.,1975,19:90–109.
[4] Kreiss R S.“Methods for the approximate solution of time dependant problems”. GARP Report,1975,13:Geneva.
[5] Adam Y. Highly accurate compact implicit methods and boundary conditions. J. Comput. Phys.,1977,24:10–22.
[6] Lele R K. Compact finite difference schemes with spectral-like resolution. J. Comput. Phys.,1992,103:16–42.
[7] Mahesh K. A family of high order finite difference schemes with good spectral resolution. J.Comput. Phys.,1998,145:332–358.
[8] Li J. High-order finite difference schemes for differential equations containing higher derivatives.Appl. Math. Comput.,2005,171:1157–1176.
[9] Carpenter M H, Gottlieb D, Abarbanel S. The stability of numerical boundary treatments forcompact high-order finite difference schemes. J. Comput. Phys.,1993,108:272–295.
[10] Kreiss H O. Stability theory for difference approximations of mixed initial boundary value prob-lems, I. Math Comput.,1968,22:703–714.
[11] Gustafsson B, Kreiss H O, Sundstrom A. Stability theory of difference approximations for mixedinitial boundary value problems, II. Math Comput.,1972,26:649–686.
[12] Chung Y M, Tucker P G. Accuracy of high-order finite difference schemes on nonuniform grids.AIAA J.,2003,41:1609–1611.
[13] Vasilyev O. High order finite difference schemes on non-uniform meshes with good conservationproperties. J. Comput. Phys.,2000,157:746–761.
[14] Shukla R K, Zhong X. Derivation of high-order compact finite difference schemes for non-uniformgrid using polynomial interpolation. J. Comput. Phys.,2005,204:404–429.
[15] Shukla R K, Tatineni M, Zhong X. Very high-order compact finite difference schemes on non-uniform grids for incomnpressible Navier-stokes equations. J. Comput. Phys.,2007,224:1064–1094.
[16] Zhong X, Tatineni M. High-order non-uniform grid schemes for numerical simulation of hyper-sonic boundary-layer stability and transition. J. Comput. Phys.,2003,190:419–458.
[17] Gamet L, Ducros F, Nicoud F, et. al. Compact finite difference schemes on non-uniform meshes.Application to direct numerical simulations of compressible flows. Int. J. Numer. Meth. Fluids,1999,29:159–191.
[18] Ma Y, Fu D. A5th order monotonicity-preserving upwind compact difference scheme. ScienceChina: Phys. Mech. Astr.,2011,54:511–522.
[19] Shah A, Yuan L. Flux-difference splitting-based upwind compact schemes for the incompressibleNavier-stokes equations. Int. J. Numer. Meth. Fluids,2009,61:552–568.
[20] Shah A, Guo H, Yuan L. A third-order upwind compact schemes on curvilinear meshes for in-compressible Navier-Stokes equations. Commun. Comput. Phys.,2009,5:712–729.
[21] Deng X, Maekawa H. Compact high-order accurate nonlinear schemes. J. Comput. Phys.,1997,130:77–91.
[22] Deng X, Zhang H. Developing high-order weighted compact nonlinear schemes. J. Comput.Phys.,2000,165:22–24.
[23] Deng X. High-order accurate dissipative weighted compact nonlinear schemes. Sci. China (Ser.A),2002,45:356–370.
[24]沈孟育，蒋莉.满足熵增原则的高精度高分辨率格式.清华大学学报(自然科学版),1999,39(4):1–5.
[25]沈孟育，蒋莉.满足“抑制波动原则”的广义紧致格式的特性及应用.应用力学学报,2003,20(2):12–16.
[26] Chu P C, Fan C. A three-point combined compact defference scheme. J. Comput. Phys.,1998,140:370–399.
[27] Chu P C, Fan C. A three-point sixth-order nonuniform combined compact difference scheme. J.Comput. Phys.,1998,148:663–674.
[28] Nihei T, Ishii K. A fast solver of the shallow water equations on a sphere using a combinedcompact difference scheme. J. Comput. Phys.,2003,187:639–659.
[29] Sengupta T K, Vijiay V V S N, Bhaumik S. A new combined stable and dispersion relationpreserving compact scheme for non-periodic problems. J. Comput. Phys.,2009,228:3048–3071.
[30] Sengupta T K, Vijiay V V S N, Bhaumik S. Further improvement and analysis of CCD scheme:Dissipation discretization and de-aliasing properties. J. Comput. Phys.,2009,228:6150–6168.
[31] Fu D, Ma Y. Analysis of super compact finite difference method and application to simulation ofvortex-shock. Int. J. Numer. Meth. Fluids,2001,36:773–805.
[32] Esfahanian V, Ghader S, Ashrafi K H. Accuracy analysis of super compact scheme in non-uniformgrid with application to parabolized stability equations. Int. J. Numer. Meth. Fluids,2004,46:485–505.
[33] Esfahanian V, Ghader S, Mohebalhojeh A R. On the use of the super compact scheme for spatialdifferencing in numerical models of the atmosphere. Q. J. R. Meteorol. Soc.,2005,131:2109–2129.
[34] Ghader S, Mohebalhojeh A R. On the spectral convergence of supercompact finite-differenceschemes for f-plane shallow-wather equations. Mon. Wea. Rev.,2009,137:2393–2406.
[35] Golshahy H, Ghader S, Ahmadi-Givi F. Accuracy assessment of the super compact and combinedcompact schemes for spatial differencing of a two-layer oceanic model: Presentation of linearinertia-gravity and Rossby waves. Ocean Model.,2011,37:49–63.
[36] Gupta M M, Manohar R P, Stephenson J W. A single cell high-order scheme for the convection-diffusion equation with variable coefficients. Int. J. Numer. Meth. Fluids,1984,4:641–651.
[37] Gupta M M, Kouatchou J, Zhang J. A compact multigrid solver for convection-diffusion equations.J. Comput. Phys.,1997,132:123–129.
[38] Spotz W F, Carey G F. A high-order compact formulation for3D poisson equation. Numer. Meth.PDEs.,1996,12:235–243.
[39] Spotz W F, Carey G F. Extension of high-order compact schemes to time-dependent problems.Numer. Meth. PDEs.,2001,17:657–672.
[40] Spotz W F, Carey G F. High-order compact scheme for the steady stream-function vorticity equa-tions. Int. J. Numer. Meth. Engng,2004,59:493–509.
[41] Lai M C. A simple compact fourth-order poisson solver on polar geometry. J. Comput. Phys.,2002,337:337–345.
[42] Kalita J C, Chhabra P. An improved (9,5) higher order compact scheme for the transient two-dimensional convection-diffusion equation. Int. J. Numer. Meth. Fluids,2006,51:703–717.
[43] Kalita J C, Dass A K, Dalal D C. A transformation-free HOC scheme for steady convection-diffusion on non-uniform grids. Int. J. Numer. Meth. Fluids,2004,44:33–53.
[44] Kalita J C, Chhabra P, Kumar S. A semi-discrete higher order compact scheme for the unsteadytwo-dimensional Schro¨dinger equation. J. Comput. Appl. Math.,2006,197:141–149.
[45] Tian Z, Dai S. High-order compact exponential finite difference methods for convection-diffusiontype problems. J. Comput. Phys,2007,220:952–974.
[46] Noye B J, Tan H H. Finite difference methods for solving the two-dimensional advection-diffusionequation. Int. J. Numer. Meth. Fluids,1989,9:75–98.
[47] Kalita J C, Dalal D C, Dass A K. A class of higher order compact schemes for the unsteadyconvection-diffusion equation with variable convection coefficients. Int. J. Numer. Meth. Fluids,2002,38:1111–1131.
[48] E W, Liu J. Essentially compact schemes for unsteady viscous incompressible flows. J. Comput.Phys.,1996,9:897–916.
[49] Li M, Tang T. A compact fourth-order finite difference scheme for unsteady viscous incompress-ible flows. J. Sci. Comput.,2001,16:29–45.
[50] Pandit S K. On the use of compact streamfunction-velocity formulation of steady Navier-Stokesequations on geometries beyond rectangular. J. Sci. Comput.,2008,36:219–242.
[51] Hassanien I A, Salama A A, Hosham H A. Fourth-order finite difference method for solvingBurger’s equation. Appl. Math. Comput.,2005,170:781–800.
[52] Liao W. An implicit fourth-order compact finite difference scheme for one-dimensional Burger’sequation. Appl. Math. Comput.,2008,206:755–764.
[53] Mohanty R K, Singh S. A new fourth order discretization for singularly perturbed two dimensionalnon-linear elliptic boundary value problems. Appl. Math. Comput.,2006,175:1400–1414.
[54] Mohanty R K, Jain M K. High accuracy difference schemes for the system of two space nonlinearparabolic differential equations with mixed derivatives and variable coefficients. J. Comput. Appl.Math.,1996,70:15–32.
[55] Mohanty R K, Arora U, Jain M K. Fourth-order approximation for the three space dimensionalcertain mildly quasi-Linear hyperbolic equation. Numer. Meth. PDEs.,2001,17:277–289.
[56] Khattar D, Singh S, Mohanty R K. A new coupled approach high accuracy numerical method forthe solution of3D non-linear biharmonic equations. J. Comput. Appl. Math.,2010,234:1312–1323.
[57] Gaitonde D, Shang J S. Optimized compact-difference-based finite-volume schemes for linearwave phenomena. J. Comput. Phys.,1997,138:617–643.
[58] Pereira J M C, Kobayashi M H, Pereira J C F. A fourth-order-accurate finite volume compactmethod for the incompressible Navier-Stokes solutions. J. Comput. Phys.,2001,167:217–243.
[59] Lacor C, Smirnova S, Baelmansb M. A finite volume formulation of compact central schemes onarbitrary structured grids. J. Comput. Phys.,2004,198:535–566.
[60] Lacor C, Smirnova S, Baelmansb M. Compact fourth-order finite volume method for numericalsolutions of Navier-Stokes equations on staggered grids. J. Comput. Phys.,2010,229:7545–7570.
[61] Gupta M M, Zhang J. High accuracy multigrid solution of the3D convection-diffusion equation.Appl. Math. Comput.,2000,113:249–274.
[62] Zhang J. Multigrid method and fourth-order compact scheme for2D poisson equation with un-equal meshsize discretization. J. Comput. Phys.,2002,179:170–179.
[63] Wang Y, Zhang J. Sixth order compact scheme combined with multigrid method and extrapolationtechnique for2D poisson equation. J. Comput. Phys.,2009,228:137–146.
[64] Golbabai A, Arabshahi M M. On the behavier of high-order compact approximations in the one-dimensional sine-Gordon equation. Phys. Scr.,2011,83:015015.
[65] Sunhaloo M S, Boojhawon R, Gopaul A, et al. On block-circulant preconditioners for high-oeder compact approximations of convection-diffusion problems. J. Comput. Appl. Math.,2010,234:1312–1323.
[66] Mayo A. Higher-order accurate implicit finite difference method for evaluating American options.Eur. J. Finance,2004,10:212–237.
[67] Liu Y, Sen M K. An implicit staggered-grid finite difference method for seismic modelling. Geo-phys. J. Int.,2009,179:459–474.
[68] Dai W, Tzou D Y. A fourth-order compact finite difference scheme for solving an N-carrier systemwith Neumann boundary conditions. Numer. Meth. PDEs.,2010,25:274–289.
[69] Dai W, Nassar R. A compact finite difference scheme for solving a three-dimensional heat trans-port equation in a thin film. Numer. Meth. PDEs.,2000,16:441–458.
[70] Peaceman D W, Jr. Rachford H H. The numerical solution of parabolic and elliptic differentialequations. J. Soc. Indust. Appl. Math.,1955,3:28–41.
[71] Jr. Douglas J, Jr. Rachford H H. On the numerical solution of heat conduction problems in two orthree space variables. Trans. Amer. Math. Soc.,1956,82:421–439.
[72] Pearcy C. On convergence of alternating direction procedures. Numer. Math.,1961,4:398–412.
[73] Jr. Douglas J. Alternating direction methods for three space varibles. Numer. Math.,1961,4:41–63.
[74] Gourlay A R, Mitchell A R. Alternating direction methods for hyperbolic systems. Numer. Math.,1966,8:137–149.
[75] Gourlay A R, Mitchell A R. The equivalence of certain alternating direction and locally one-dimensional difference methods. SIAM J. Numer. Anal,1969,6:37–46.
[76] McKee S, Mitchell A R. Alternating direction methods for parabolic equations in two spacedimensions with a mixed derivative. The Comput. J.,1970,13:81–86.
[77] McKee S, Wall D P, Wilson S K. An alternating direction implicit scheme for parabolic equationwith mixed derivative and convection terms. J. Comput. Phys.,1996,126:64–76.
[78] Jr. Douglas J. Alternating direction iteration for mildly nonlinear elliptic difference equations.Numer. Math.,1961,3:92–98.
[79] Jr. Douglas J, Gunn J E. A general formulation of alternating direction method: Part I. Parabolicand hyperbolic problem. Numer. Math.,1964,6:428–453.
[80] Fairweather G, Mitchell A R. A new computational procedure for A.D.I. methods. SIAM J.Numer. Anal.,1967,4:163–170.
[81] Mitchell A R, Fairweather G. Improved forms of the alternating direction methods of Douglas,Peaceman, and Rachford for solving parabolic and elliptic equations. Numer. Math.,1964,6:428–453.
[82] In’T Hout K, Welfert B D. Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms,. Appl. Numer. Math.,2009,59:677–692.
[83] Beam R M, Warming R F. An implicit finite difference algorithm for hyperbolic systems inconservation-law form. J. Comput. Phys.,1976,22:87–110.
[84] Warming R F, Beam R M. An extension of A-stability to alternating direction implicit method.BIT Numer. Math.,1979,19:1111–1131.
[85] Zhang Z, Deng D. A new alternating-direction finite element method for hyperbolic equation.Numer. Meth. PDEs.,2007,23:1530–1559.
[86] Yang M. Two-time level ADI finite volume method for a class of second-order hyperbolic prob-lems. Appl. Math. Comput.,2010,215:3239–3248.
[87] Bialeck B, Fernandes R I. An orthogonal spline collocation alternating derection implicit Crank-Nicolson method for linear parabolic problems on rectangles. SIAM Numer. Anal.,1999,36:1414–1434.
[88] Karaa S, Zhang J. High order ADI method for solving unsteady convection-diffusion problems. J.Comput. Phys.,2004,198:1–9.
[89] Karaa S. An accurate LOD scheme for two-dimensional parabolic problems. Appl. Math. Com-put.,2005,170:886–894.
[90] You D. A high-order Pade′ADI method for unstesdy convection-diffusion equations. J. Comput.Phys.,2006,214:1–11.
[91] Tian Z, Ge Y. A fourth-order compact ADI method for solving two-dimensional unsteadyconvection-diffusion problems. J. Comput. Appl. Math.,2007,198:268–286.
[92] Karaa S. A hybrid PadéADI scheme of higher-order for convection-diffusion problems. Int. J.Numer. Meth. Fluids,2010,64:532–548.
[93] Tian Z. A rational high-order compact ADI method for unsteady convection-diffusion equations.Comput. Phys. Commun.,2011,182:649–662.
[94] Gao Z, Xie S. Fourth-order alternating direction implicit compact finite difference schemes fortwo-dimensional schro¨dinger equations. Appl. Numer. Math.,2011,61:593–614.
[95] Tian Z，Xu P. High-order compact ADI (HOC-ADI) method for solving unsteady2D schro¨dingerequation. Comput. Phys. Commun.,2010,181:861–868.
[96] Xu Y, Zhang L. Alternating direction implicit method for solving two-dimensional cubic nonlinearschro¨dinge equation. Comput. Phys. Commun.,2012,183:1082–1093.
[97] Karaa S. Unconditionally stable ADI scheme of high-order for linear hyperbolic equations. Int. J.Comput. Math.,2010,87:3030–3038.
[98] Liu J, Tang K. A new unconditionally stable ADI compact scheme for the two-space-dimensionallinear hyperbolic equation. Int. J. Comput. Math.,2010,87:2259–2267.
[99] Cui M. High order compact alternating direction implicit method for the generalized sine-Gordonequation. J. Comput. Appl. Math.,2010,235:837–849.
[100] Dai W, Nassar R. Compact ADI method for solving parabolic differential equations. Numer. Meth.PDEs.,2002,18:129–142.
[101] Li J, Chen Y, Liu G. High-order compact ADI Methods for parabolic equations. Comput. Math.Appl.,2006,52:1343–1356.
[102] Liao W, Zhu J, Khaliq A Q M. An efficient high-order algorithm for solving systems of reaction-diffusion equations. Numer. Meth. PDEs.,2002,18:340–354.
[103] Gu Y, Liao W, Zhu J. An efficient high-order algorithm for solving systems of3-D reaction-diffusion equations. J. Comput. Appl. Math.,2003,155:1–17.
[104]孙志忠，李雪玲.反应扩散方程的紧交替方向的差分方法.计算数学,2005,27(2):209–224.
[105] Liao H, Sun Z. Maximum norm error bounds of ADI and compact ADI methods for solvingparabolic equations. Numer. Meth. PDEs.,2010,26:37–60.
[106] Qin J, Wang T. A compact locally one-dimensional finite difference method for nonhomogeneousparabolic differential equations. Int. J. Numer. Meth. Biomed. Engng.,2011,27:128–142.
[107] Wang Y. Error and extrapolation of a compact LOD method for parabolic differential equations.J. Comput. Appl. Math.,2011,235:1367–1382.
[108] Jordan P M, Puri A. Digital signal propagation in dispersive media. J. Appl. Phys.,1999,85:1272–1282.
[109] Jordan P M, Meyer M R, Puri A. Causal implications of viscous damping in compressible fluidflows. Phys. Rev. E,2000,62:7918–7926.
[110] Orsingher E. A planar random motion governed by the two-dimensional telegraph equation. J.Appl. Prob.,1986,23:385–397.
[111] Mawhin J, Ortega R, Robles-Perez A M. Maximum principles for bounded solutions of the tele-graph equation in space dimensions two and three and applications. J. Differential Equations,2005,208:42–63.
[112] Ortega R, Robles-Perez A M. A maximum principle for periodic solutions of the telegraph equa-tion. J. Math. Anal. Appl.,1998,221:625–651.
[113] Chen J, Liu F, Anhb V. Analytical solution for the time-fractional telegraph equation by themethod of separating variables. J. Math. Anal. Appl.,2008,338:1364–1377.
[114] Biazar J. Analytic solution for telegraph equation by differential transform. Phys. Lett. A,2010,374:2904–2906.
[115] Dehghan M, Shokri A. A numerical method for solving the hyperbolic telegraph equation. Numer.Meth. PDEs.,2008,24:1080–1093.
[116] Dehghan M, Mohebbi A. High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation. Numer. Meth. PDEs.,2009,25:232–243.
[117] Dehghan M, Ghesmati A. Combination of meshless local weak and strong (MLWS) forms to solvethe two dimensional hyperbolic telegraph equation. Eng. Anal. Bound. Elem.,2010,34:324–336.
[118] Yousefi S A. Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equa-tion. Numer. Meth. PDEs.,2010,26:535–543.
[119] Rashidinia J, Mohammadi R, Jalilian R. Spline methods for the solution of hyperbolic equationwith variable coefficients. Numer. Meth. PDEs.,2007,23:1411–1419.
[120] Evans D J, Bulut H. The numerical solution of the telegraph equation by the alternating groupexplicit (AGE) method. Int. J. Comput. Math.,2003,80:1289–1297.
[121] Sun H, Qin M, Wang Y, et al. Multi-symplectic Birkhoffian structure for PDEs with dissipationterms. Phys. Lett. A,2010,374:2410–2416.
[122] Mickens R E, Jordan P M. A positivity-preserving non-standard finite difference scheme for thedamped wave equation. Numer. Meth. PDEs.,2004,20:639–649.
[123] Ding H, Zhang Y. A new fourth-order compact finite difference scheme for the two-dimensionalsecond-order hyperbolic equation. J. Comput. Appl. Math.,2009,230:626–632.
[124] Mohanty R K, Jain M K. An unconditionally stable alternating direction implicit scheme for thetwo space dimensional linear hyperbolic equation. Numer. Meth. PDEs.,2001,17:684–688.
[125] Mohanty R K, Jain M K, Arora U. An unconditionally stable ADI method for the linear hyperbolicequation in three space dimensions. Int. J. Comput. Math.,2002,79:133–142.
[126] Mohanty R K. An operator splitting method for an unconditionally stable difference scheme fora linear hyperbolic equation with variable coefficients in two space dimensions. Appl. Math.Comput.,2004,152:799–806.
[127] Mohanty R K. An operator splitting technique for an unconditionally stable difference methodfor a linear three space dimensional hyperbolic equation with variable coefficients. Appl. Math.Comput.,2005,162:549–557.
[128] Mohanty R K. New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations. Int. J. Comput. Math.,2009,86:2061–2071.
[129] Liu L, Liu H. Quartic spline methods for solving one-dimensional telegraphic equations. Appl.Math. Comput.,2010,216:951–958.
[130] Gulevich D R, Kusmartsev F V, Savel’ev S, et al. Shape and wobbling wave excitations in Joseph-son junctions: exact solutions of the (2+1)-dimensional sine-Gordon model. Phys. Rev. B,2009,80:0945093.
[131] Leibbrandt G. New exact solutions of the classical sine-Gordon equation in2+1and3+1dimen-sions. Phys. Rev. Lett,1978,41:435–438.
[132] Hosseinzadeh H, Jafari H, Roohani M. Application of Laplace decomposition method for solvingKlein-Gordon equation. World Appl. Sci. J.,2010,8:809–813.
[133] Bratsos A G. The solution of the two-dimensional sine-Gordon equation using the method of lines.J. Comput. Appl. Math.,2007,206:251–277.
[134] Dehghan M, Mohebbi A, Asgari Z. Fourth-order compact solution of the nonlinear Klein-Gordonequation. Numer. Algor.,2009,52:523–540.
[135] Djidjeli K, Price W G, Twizell E H. Numerical solutions of a damped sine-Gordon equation intwo space variables. J. Engrg. Math.,1995,29:347–369.
[136] Zheng C. Numerical solution to the sine-Gordon equation defined on the whole real axis. SIAMJ. Sci. Comput.,2007,29:2494–2506.
[137] Sheng Q, Khaliq A Q M, Voss D A. Numerical simulation of two-dimensional sine-Gordon soli-tons via a split cosine scheme. Math. Comput. Simul.,2005,68:355–373.
[138] Khuri S A, Sayfy A. A spline collocation approach for the numerical solution of generalizednonlinear Klein-Gordon equation. Appl. Math. Comput.,2010,187:1047–1056.
[139] Rashidinia J, Mohammadi R. Tension spline approach for the numerical solution of nonlinearKlein-Gordon equation. Comput. Phys. Commun.,2010,181:78–91.
[140] Rashidinia J, Ghasemi M, Jalilian R. Numerical solution of the nonlinear Klein-Gordon equation.J. Comput. Appl. Math.,2007,206:251–277.
[141]朱玲，张志跃.一类非线性发展方程的全离散有限体积元方法及其分析.计算力学学报,2008,25(3):304–309.
[142] Liu Y, Yu T. Blow-up of the nerve conduction equation. Acta Math. Appl. Sinica,1995,18:264–271.
[143] Moll V, Rorencrans S I. Calculation of the threshold surface for nerve equation. SIAM Appl.Math.,1990,50:1419–1441.
[144] Zhang Z. The finite element numerical analysis for a class of nonlinear evolution equations. Appl.Math. Comput.,2005,166:489–500.
[145]张志跃.一类非线性发展方程的交替分段显隐并行数值方法.计算力学学报,2002,19(2):154–158.
[146] Zhang Z. The multistep finite difference fractional steps method for a class of viscous waveequations. Math. Meth. Appl. Sci.,2011,34:442–454.
[147] Deng D, Zhang Z. A new high-order algorithm for a class of nonlinear evolution equation. J. Phys.A: Math. Theor.,2008,41:015202.
[148] Zhang Z, Deng D. A fourth-order finite difference solver for nerve conduction equation in rectan-gular domains. J. Math. Phys.,2008,49:043508.
[149] Dai W, Nassar R. An unconditionally stable finite difference scheme for solving a3D heat transportequation in sub-microscale thin film. J. Comput. Appl. Math.,2002,145:247–260.
[150] Zhang Z. An economical difference scheme for heat transport equation at the microscale. Numer.Meth. PDEs.,2004,20:855–863.
[151]王同科.一类二维粘性波动方程的交替方向有限体积元方法.数值计算与计算机应用,2010,31(1):64–75.
[152] Verwer J G. Runge-Kutta methods and viscous wave equations. Numer. Math.,2009,112:485–507.
[153] Malek A, Momeni-Masuleh S H. A mixed collocation-finite difference method for3D microscopicheat transport problems. J. Comput. Appl. Math.,2008,217:137–147.
[154] Lim H, Kim S, Jr. Douglas J. Numerical methods for viscous and nonviscous wave equations.Appl. Numer. Math.,2007,57:194–212.
[155] Karaa S. Approximate factorization for a viscous wave equation. Computing,2010,89:199–215.
[156] Karaa S. Error estimates for finite element approximates of a viscous wave equation. Numer.Func. Anal. Opt.,2011,32:750–767.
[157] Liao H, Sun Z. Maximum norm error estimates of efficient difference schemes for second-orderwave equations. J. Comput. Appl. Math.,2011,235:2217–2233.
[2] Adam Y. A hermitian finite difference method for the solution of parabolic equations. Comput.Math. Appl.,1975,1:393–406.
[3] Hirsch R S. High order accurate difference solutions of fluid mechanics problems by a compactdifferencing technique. J. Comput. Phys.,1975,19:90–109.
[4] Kreiss R S.“Methods for the approximate solution of time dependant problems”. GARP Report,1975,13:Geneva.
[5] Adam Y. Highly accurate compact implicit methods and boundary conditions. J. Comput. Phys.,1977,24:10–22.
[6] Lele R K. Compact finite difference schemes with spectral-like resolution. J. Comput. Phys.,1992,103:16–42.
[7] Mahesh K. A family of high order finite difference schemes with good spectral resolution. J.Comput. Phys.,1998,145:332–358.
[8] Li J. High-order finite difference schemes for differential equations containing higher derivatives.Appl. Math. Comput.,2005,171:1157–1176.
[9] Carpenter M H, Gottlieb D, Abarbanel S. The stability of numerical boundary treatments forcompact high-order finite difference schemes. J. Comput. Phys.,1993,108:272–295.
[10] Kreiss H O. Stability theory for difference approximations of mixed initial boundary value prob-lems, I. Math Comput.,1968,22:703–714.
[11] Gustafsson B, Kreiss H O, Sundstrom A. Stability theory of difference approximations for mixedinitial boundary value problems, II. Math Comput.,1972,26:649–686.
[12] Chung Y M, Tucker P G. Accuracy of high-order finite difference schemes on nonuniform grids.AIAA J.,2003,41:1609–1611.
[13] Vasilyev O. High order finite difference schemes on non-uniform meshes with good conservationproperties. J. Comput. Phys.,2000,157:746–761.
[14] Shukla R K, Zhong X. Derivation of high-order compact finite difference schemes for non-uniformgrid using polynomial interpolation. J. Comput. Phys.,2005,204:404–429.
[15] Shukla R K, Tatineni M, Zhong X. Very high-order compact finite difference schemes on non-uniform grids for incomnpressible Navier-stokes equations. J. Comput. Phys.,2007,224:1064–1094.
[16] Zhong X, Tatineni M. High-order non-uniform grid schemes for numerical simulation of hyper-sonic boundary-layer stability and transition. J. Comput. Phys.,2003,190:419–458.
[17] Gamet L, Ducros F, Nicoud F, et. al. Compact finite difference schemes on non-uniform meshes.Application to direct numerical simulations of compressible flows. Int. J. Numer. Meth. Fluids,1999,29:159–191.
[18] Ma Y, Fu D. A5th order monotonicity-preserving upwind compact difference scheme. ScienceChina: Phys. Mech. Astr.,2011,54:511–522.
[19] Shah A, Yuan L. Flux-difference splitting-based upwind compact schemes for the incompressibleNavier-stokes equations. Int. J. Numer. Meth. Fluids,2009,61:552–568.
[20] Shah A, Guo H, Yuan L. A third-order upwind compact schemes on curvilinear meshes for in-compressible Navier-Stokes equations. Commun. Comput. Phys.,2009,5:712–729.
[21] Deng X, Maekawa H. Compact high-order accurate nonlinear schemes. J. Comput. Phys.,1997,130:77–91.
[22] Deng X, Zhang H. Developing high-order weighted compact nonlinear schemes. J. Comput.Phys.,2000,165:22–24.
[23] Deng X. High-order accurate dissipative weighted compact nonlinear schemes. Sci. China (Ser.A),2002,45:356–370.
[24]沈孟育，蒋莉.满足熵增原则的高精度高分辨率格式.清华大学学报(自然科学版),1999,39(4):1–5.
[25]沈孟育，蒋莉.满足“抑制波动原则”的广义紧致格式的特性及应用.应用力学学报,2003,20(2):12–16.
[26] Chu P C, Fan C. A three-point combined compact defference scheme. J. Comput. Phys.,1998,140:370–399.
[27] Chu P C, Fan C. A three-point sixth-order nonuniform combined compact difference scheme. J.Comput. Phys.,1998,148:663–674.
[28] Nihei T, Ishii K. A fast solver of the shallow water equations on a sphere using a combinedcompact difference scheme. J. Comput. Phys.,2003,187:639–659.
[29] Sengupta T K, Vijiay V V S N, Bhaumik S. A new combined stable and dispersion relationpreserving compact scheme for non-periodic problems. J. Comput. Phys.,2009,228:3048–3071.
[30] Sengupta T K, Vijiay V V S N, Bhaumik S. Further improvement and analysis of CCD scheme:Dissipation discretization and de-aliasing properties. J. Comput. Phys.,2009,228:6150–6168.
[31] Fu D, Ma Y. Analysis of super compact finite difference method and application to simulation ofvortex-shock. Int. J. Numer. Meth. Fluids,2001,36:773–805.
[32] Esfahanian V, Ghader S, Ashrafi K H. Accuracy analysis of super compact scheme in non-uniformgrid with application to parabolized stability equations. Int. J. Numer. Meth. Fluids,2004,46:485–505.
[33] Esfahanian V, Ghader S, Mohebalhojeh A R. On the use of the super compact scheme for spatialdifferencing in numerical models of the atmosphere. Q. J. R. Meteorol. Soc.,2005,131:2109–2129.
[34] Ghader S, Mohebalhojeh A R. On the spectral convergence of supercompact finite-differenceschemes for f-plane shallow-wather equations. Mon. Wea. Rev.,2009,137:2393–2406.
[35] Golshahy H, Ghader S, Ahmadi-Givi F. Accuracy assessment of the super compact and combinedcompact schemes for spatial differencing of a two-layer oceanic model: Presentation of linearinertia-gravity and Rossby waves. Ocean Model.,2011,37:49–63.
[36] Gupta M M, Manohar R P, Stephenson J W. A single cell high-order scheme for the convection-diffusion equation with variable coefficients. Int. J. Numer. Meth. Fluids,1984,4:641–651.
[37] Gupta M M, Kouatchou J, Zhang J. A compact multigrid solver for convection-diffusion equations.J. Comput. Phys.,1997,132:123–129.
[38] Spotz W F, Carey G F. A high-order compact formulation for3D poisson equation. Numer. Meth.PDEs.,1996,12:235–243.
[39] Spotz W F, Carey G F. Extension of high-order compact schemes to time-dependent problems.Numer. Meth. PDEs.,2001,17:657–672.
[40] Spotz W F, Carey G F. High-order compact scheme for the steady stream-function vorticity equa-tions. Int. J. Numer. Meth. Engng,2004,59:493–509.
[41] Lai M C. A simple compact fourth-order poisson solver on polar geometry. J. Comput. Phys.,2002,337:337–345.
[42] Kalita J C, Chhabra P. An improved (9,5) higher order compact scheme for the transient two-dimensional convection-diffusion equation. Int. J. Numer. Meth. Fluids,2006,51:703–717.
[43] Kalita J C, Dass A K, Dalal D C. A transformation-free HOC scheme for steady convection-diffusion on non-uniform grids. Int. J. Numer. Meth. Fluids,2004,44:33–53.
[44] Kalita J C, Chhabra P, Kumar S. A semi-discrete higher order compact scheme for the unsteadytwo-dimensional Schro¨dinger equation. J. Comput. Appl. Math.,2006,197:141–149.
[45] Tian Z, Dai S. High-order compact exponential finite difference methods for convection-diffusiontype problems. J. Comput. Phys,2007,220:952–974.
[46] Noye B J, Tan H H. Finite difference methods for solving the two-dimensional advection-diffusionequation. Int. J. Numer. Meth. Fluids,1989,9:75–98.
[47] Kalita J C, Dalal D C, Dass A K. A class of higher order compact schemes for the unsteadyconvection-diffusion equation with variable convection coefficients. Int. J. Numer. Meth. Fluids,2002,38:1111–1131.
[48] E W, Liu J. Essentially compact schemes for unsteady viscous incompressible flows. J. Comput.Phys.,1996,9:897–916.
[49] Li M, Tang T. A compact fourth-order finite difference scheme for unsteady viscous incompress-ible flows. J. Sci. Comput.,2001,16:29–45.
[50] Pandit S K. On the use of compact streamfunction-velocity formulation of steady Navier-Stokesequations on geometries beyond rectangular. J. Sci. Comput.,2008,36:219–242.
[51] Hassanien I A, Salama A A, Hosham H A. Fourth-order finite difference method for solvingBurger’s equation. Appl. Math. Comput.,2005,170:781–800.
[52] Liao W. An implicit fourth-order compact finite difference scheme for one-dimensional Burger’sequation. Appl. Math. Comput.,2008,206:755–764.
[53] Mohanty R K, Singh S. A new fourth order discretization for singularly perturbed two dimensionalnon-linear elliptic boundary value problems. Appl. Math. Comput.,2006,175:1400–1414.
[54] Mohanty R K, Jain M K. High accuracy difference schemes for the system of two space nonlinearparabolic differential equations with mixed derivatives and variable coefficients. J. Comput. Appl.Math.,1996,70:15–32.
[55] Mohanty R K, Arora U, Jain M K. Fourth-order approximation for the three space dimensionalcertain mildly quasi-Linear hyperbolic equation. Numer. Meth. PDEs.,2001,17:277–289.
[56] Khattar D, Singh S, Mohanty R K. A new coupled approach high accuracy numerical method forthe solution of3D non-linear biharmonic equations. J. Comput. Appl. Math.,2010,234:1312–1323.
[57] Gaitonde D, Shang J S. Optimized compact-difference-based finite-volume schemes for linearwave phenomena. J. Comput. Phys.,1997,138:617–643.
[58] Pereira J M C, Kobayashi M H, Pereira J C F. A fourth-order-accurate finite volume compactmethod for the incompressible Navier-Stokes solutions. J. Comput. Phys.,2001,167:217–243.
[59] Lacor C, Smirnova S, Baelmansb M. A finite volume formulation of compact central schemes onarbitrary structured grids. J. Comput. Phys.,2004,198:535–566.
[60] Lacor C, Smirnova S, Baelmansb M. Compact fourth-order finite volume method for numericalsolutions of Navier-Stokes equations on staggered grids. J. Comput. Phys.,2010,229:7545–7570.
[61] Gupta M M, Zhang J. High accuracy multigrid solution of the3D convection-diffusion equation.Appl. Math. Comput.,2000,113:249–274.
[62] Zhang J. Multigrid method and fourth-order compact scheme for2D poisson equation with un-equal meshsize discretization. J. Comput. Phys.,2002,179:170–179.
[63] Wang Y, Zhang J. Sixth order compact scheme combined with multigrid method and extrapolationtechnique for2D poisson equation. J. Comput. Phys.,2009,228:137–146.
[64] Golbabai A, Arabshahi M M. On the behavier of high-order compact approximations in the one-dimensional sine-Gordon equation. Phys. Scr.,2011,83:015015.
[65] Sunhaloo M S, Boojhawon R, Gopaul A, et al. On block-circulant preconditioners for high-oeder compact approximations of convection-diffusion problems. J. Comput. Appl. Math.,2010,234:1312–1323.
[66] Mayo A. Higher-order accurate implicit finite difference method for evaluating American options.Eur. J. Finance,2004,10:212–237.
[67] Liu Y, Sen M K. An implicit staggered-grid finite difference method for seismic modelling. Geo-phys. J. Int.,2009,179:459–474.
[68] Dai W, Tzou D Y. A fourth-order compact finite difference scheme for solving an N-carrier systemwith Neumann boundary conditions. Numer. Meth. PDEs.,2010,25:274–289.
[69] Dai W, Nassar R. A compact finite difference scheme for solving a three-dimensional heat trans-port equation in a thin film. Numer. Meth. PDEs.,2000,16:441–458.
[70] Peaceman D W, Jr. Rachford H H. The numerical solution of parabolic and elliptic differentialequations. J. Soc. Indust. Appl. Math.,1955,3:28–41.
[71] Jr. Douglas J, Jr. Rachford H H. On the numerical solution of heat conduction problems in two orthree space variables. Trans. Amer. Math. Soc.,1956,82:421–439.
[72] Pearcy C. On convergence of alternating direction procedures. Numer. Math.,1961,4:398–412.
[73] Jr. Douglas J. Alternating direction methods for three space varibles. Numer. Math.,1961,4:41–63.
[74] Gourlay A R, Mitchell A R. Alternating direction methods for hyperbolic systems. Numer. Math.,1966,8:137–149.
[75] Gourlay A R, Mitchell A R. The equivalence of certain alternating direction and locally one-dimensional difference methods. SIAM J. Numer. Anal,1969,6:37–46.
[76] McKee S, Mitchell A R. Alternating direction methods for parabolic equations in two spacedimensions with a mixed derivative. The Comput. J.,1970,13:81–86.
[77] McKee S, Wall D P, Wilson S K. An alternating direction implicit scheme for parabolic equationwith mixed derivative and convection terms. J. Comput. Phys.,1996,126:64–76.
[78] Jr. Douglas J. Alternating direction iteration for mildly nonlinear elliptic difference equations.Numer. Math.,1961,3:92–98.
[79] Jr. Douglas J, Gunn J E. A general formulation of alternating direction method: Part I. Parabolicand hyperbolic problem. Numer. Math.,1964,6:428–453.
[80] Fairweather G, Mitchell A R. A new computational procedure for A.D.I. methods. SIAM J.Numer. Anal.,1967,4:163–170.
[81] Mitchell A R, Fairweather G. Improved forms of the alternating direction methods of Douglas,Peaceman, and Rachford for solving parabolic and elliptic equations. Numer. Math.,1964,6:428–453.
[82] In’T Hout K, Welfert B D. Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms,. Appl. Numer. Math.,2009,59:677–692.
[83] Beam R M, Warming R F. An implicit finite difference algorithm for hyperbolic systems inconservation-law form. J. Comput. Phys.,1976,22:87–110.
[84] Warming R F, Beam R M. An extension of A-stability to alternating direction implicit method.BIT Numer. Math.,1979,19:1111–1131.
[85] Zhang Z, Deng D. A new alternating-direction finite element method for hyperbolic equation.Numer. Meth. PDEs.,2007,23:1530–1559.
[86] Yang M. Two-time level ADI finite volume method for a class of second-order hyperbolic prob-lems. Appl. Math. Comput.,2010,215:3239–3248.
[87] Bialeck B, Fernandes R I. An orthogonal spline collocation alternating derection implicit Crank-Nicolson method for linear parabolic problems on rectangles. SIAM Numer. Anal.,1999,36:1414–1434.
[88] Karaa S, Zhang J. High order ADI method for solving unsteady convection-diffusion problems. J.Comput. Phys.,2004,198:1–9.
[89] Karaa S. An accurate LOD scheme for two-dimensional parabolic problems. Appl. Math. Com-put.,2005,170:886–894.
[90] You D. A high-order Pade′ADI method for unstesdy convection-diffusion equations. J. Comput.Phys.,2006,214:1–11.
[91] Tian Z, Ge Y. A fourth-order compact ADI method for solving two-dimensional unsteadyconvection-diffusion problems. J. Comput. Appl. Math.,2007,198:268–286.
[92] Karaa S. A hybrid PadéADI scheme of higher-order for convection-diffusion problems. Int. J.Numer. Meth. Fluids,2010,64:532–548.
[93] Tian Z. A rational high-order compact ADI method for unsteady convection-diffusion equations.Comput. Phys. Commun.,2011,182:649–662.
[94] Gao Z, Xie S. Fourth-order alternating direction implicit compact finite difference schemes fortwo-dimensional schro¨dinger equations. Appl. Numer. Math.,2011,61:593–614.
[95] Tian Z，Xu P. High-order compact ADI (HOC-ADI) method for solving unsteady2D schro¨dingerequation. Comput. Phys. Commun.,2010,181:861–868.
[96] Xu Y, Zhang L. Alternating direction implicit method for solving two-dimensional cubic nonlinearschro¨dinge equation. Comput. Phys. Commun.,2012,183:1082–1093.
[97] Karaa S. Unconditionally stable ADI scheme of high-order for linear hyperbolic equations. Int. J.Comput. Math.,2010,87:3030–3038.
[98] Liu J, Tang K. A new unconditionally stable ADI compact scheme for the two-space-dimensionallinear hyperbolic equation. Int. J. Comput. Math.,2010,87:2259–2267.
[99] Cui M. High order compact alternating direction implicit method for the generalized sine-Gordonequation. J. Comput. Appl. Math.,2010,235:837–849.
[100] Dai W, Nassar R. Compact ADI method for solving parabolic differential equations. Numer. Meth.PDEs.,2002,18:129–142.
[101] Li J, Chen Y, Liu G. High-order compact ADI Methods for parabolic equations. Comput. Math.Appl.,2006,52:1343–1356.
[102] Liao W, Zhu J, Khaliq A Q M. An efficient high-order algorithm for solving systems of reaction-diffusion equations. Numer. Meth. PDEs.,2002,18:340–354.
[103] Gu Y, Liao W, Zhu J. An efficient high-order algorithm for solving systems of3-D reaction-diffusion equations. J. Comput. Appl. Math.,2003,155:1–17.
[104]孙志忠，李雪玲.反应扩散方程的紧交替方向的差分方法.计算数学,2005,27(2):209–224.
[105] Liao H, Sun Z. Maximum norm error bounds of ADI and compact ADI methods for solvingparabolic equations. Numer. Meth. PDEs.,2010,26:37–60.
[106] Qin J, Wang T. A compact locally one-dimensional finite difference method for nonhomogeneousparabolic differential equations. Int. J. Numer. Meth. Biomed. Engng.,2011,27:128–142.
[107] Wang Y. Error and extrapolation of a compact LOD method for parabolic differential equations.J. Comput. Appl. Math.,2011,235:1367–1382.
[108] Jordan P M, Puri A. Digital signal propagation in dispersive media. J. Appl. Phys.,1999,85:1272–1282.
[109] Jordan P M, Meyer M R, Puri A. Causal implications of viscous damping in compressible fluidflows. Phys. Rev. E,2000,62:7918–7926.
[110] Orsingher E. A planar random motion governed by the two-dimensional telegraph equation. J.Appl. Prob.,1986,23:385–397.
[111] Mawhin J, Ortega R, Robles-Perez A M. Maximum principles for bounded solutions of the tele-graph equation in space dimensions two and three and applications. J. Differential Equations,2005,208:42–63.
[112] Ortega R, Robles-Perez A M. A maximum principle for periodic solutions of the telegraph equa-tion. J. Math. Anal. Appl.,1998,221:625–651.
[113] Chen J, Liu F, Anhb V. Analytical solution for the time-fractional telegraph equation by themethod of separating variables. J. Math. Anal. Appl.,2008,338:1364–1377.
[114] Biazar J. Analytic solution for telegraph equation by differential transform. Phys. Lett. A,2010,374:2904–2906.
[115] Dehghan M, Shokri A. A numerical method for solving the hyperbolic telegraph equation. Numer.Meth. PDEs.,2008,24:1080–1093.
[116] Dehghan M, Mohebbi A. High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation. Numer. Meth. PDEs.,2009,25:232–243.
[117] Dehghan M, Ghesmati A. Combination of meshless local weak and strong (MLWS) forms to solvethe two dimensional hyperbolic telegraph equation. Eng. Anal. Bound. Elem.,2010,34:324–336.
[118] Yousefi S A. Legendre multiwavelet Galerkin method for solving the hyperbolic telegraph equa-tion. Numer. Meth. PDEs.,2010,26:535–543.
[119] Rashidinia J, Mohammadi R, Jalilian R. Spline methods for the solution of hyperbolic equationwith variable coefficients. Numer. Meth. PDEs.,2007,23:1411–1419.
[120] Evans D J, Bulut H. The numerical solution of the telegraph equation by the alternating groupexplicit (AGE) method. Int. J. Comput. Math.,2003,80:1289–1297.
[121] Sun H, Qin M, Wang Y, et al. Multi-symplectic Birkhoffian structure for PDEs with dissipationterms. Phys. Lett. A,2010,374:2410–2416.
[122] Mickens R E, Jordan P M. A positivity-preserving non-standard finite difference scheme for thedamped wave equation. Numer. Meth. PDEs.,2004,20:639–649.
[123] Ding H, Zhang Y. A new fourth-order compact finite difference scheme for the two-dimensionalsecond-order hyperbolic equation. J. Comput. Appl. Math.,2009,230:626–632.
[124] Mohanty R K, Jain M K. An unconditionally stable alternating direction implicit scheme for thetwo space dimensional linear hyperbolic equation. Numer. Meth. PDEs.,2001,17:684–688.
[125] Mohanty R K, Jain M K, Arora U. An unconditionally stable ADI method for the linear hyperbolicequation in three space dimensions. Int. J. Comput. Math.,2002,79:133–142.
[126] Mohanty R K. An operator splitting method for an unconditionally stable difference scheme fora linear hyperbolic equation with variable coefficients in two space dimensions. Appl. Math.Comput.,2004,152:799–806.
[127] Mohanty R K. An operator splitting technique for an unconditionally stable difference methodfor a linear three space dimensional hyperbolic equation with variable coefficients. Appl. Math.Comput.,2005,162:549–557.
[128] Mohanty R K. New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations. Int. J. Comput. Math.,2009,86:2061–2071.
[129] Liu L, Liu H. Quartic spline methods for solving one-dimensional telegraphic equations. Appl.Math. Comput.,2010,216:951–958.
[130] Gulevich D R, Kusmartsev F V, Savel’ev S, et al. Shape and wobbling wave excitations in Joseph-son junctions: exact solutions of the (2+1)-dimensional sine-Gordon model. Phys. Rev. B,2009,80:0945093.
[131] Leibbrandt G. New exact solutions of the classical sine-Gordon equation in2+1and3+1dimen-sions. Phys. Rev. Lett,1978,41:435–438.
[132] Hosseinzadeh H, Jafari H, Roohani M. Application of Laplace decomposition method for solvingKlein-Gordon equation. World Appl. Sci. J.,2010,8:809–813.
[133] Bratsos A G. The solution of the two-dimensional sine-Gordon equation using the method of lines.J. Comput. Appl. Math.,2007,206:251–277.
[134] Dehghan M, Mohebbi A, Asgari Z. Fourth-order compact solution of the nonlinear Klein-Gordonequation. Numer. Algor.,2009,52:523–540.
[135] Djidjeli K, Price W G, Twizell E H. Numerical solutions of a damped sine-Gordon equation intwo space variables. J. Engrg. Math.,1995,29:347–369.
[136] Zheng C. Numerical solution to the sine-Gordon equation defined on the whole real axis. SIAMJ. Sci. Comput.,2007,29:2494–2506.
[137] Sheng Q, Khaliq A Q M, Voss D A. Numerical simulation of two-dimensional sine-Gordon soli-tons via a split cosine scheme. Math. Comput. Simul.,2005,68:355–373.
[138] Khuri S A, Sayfy A. A spline collocation approach for the numerical solution of generalizednonlinear Klein-Gordon equation. Appl. Math. Comput.,2010,187:1047–1056.
[139] Rashidinia J, Mohammadi R. Tension spline approach for the numerical solution of nonlinearKlein-Gordon equation. Comput. Phys. Commun.,2010,181:78–91.
[140] Rashidinia J, Ghasemi M, Jalilian R. Numerical solution of the nonlinear Klein-Gordon equation.J. Comput. Appl. Math.,2007,206:251–277.
[141]朱玲，张志跃.一类非线性发展方程的全离散有限体积元方法及其分析.计算力学学报,2008,25(3):304–309.
[142] Liu Y, Yu T. Blow-up of the nerve conduction equation. Acta Math. Appl. Sinica,1995,18:264–271.
[143] Moll V, Rorencrans S I. Calculation of the threshold surface for nerve equation. SIAM Appl.Math.,1990,50:1419–1441.
[144] Zhang Z. The finite element numerical analysis for a class of nonlinear evolution equations. Appl.Math. Comput.,2005,166:489–500.
[145]张志跃.一类非线性发展方程的交替分段显隐并行数值方法.计算力学学报,2002,19(2):154–158.
[146] Zhang Z. The multistep finite difference fractional steps method for a class of viscous waveequations. Math. Meth. Appl. Sci.,2011,34:442–454.
[147] Deng D, Zhang Z. A new high-order algorithm for a class of nonlinear evolution equation. J. Phys.A: Math. Theor.,2008,41:015202.
[148] Zhang Z, Deng D. A fourth-order finite difference solver for nerve conduction equation in rectan-gular domains. J. Math. Phys.,2008,49:043508.
[149] Dai W, Nassar R. An unconditionally stable finite difference scheme for solving a3D heat transportequation in sub-microscale thin film. J. Comput. Appl. Math.,2002,145:247–260.
[150] Zhang Z. An economical difference scheme for heat transport equation at the microscale. Numer.Meth. PDEs.,2004,20:855–863.
[151]王同科.一类二维粘性波动方程的交替方向有限体积元方法.数值计算与计算机应用,2010,31(1):64–75.
[152] Verwer J G. Runge-Kutta methods and viscous wave equations. Numer. Math.,2009,112:485–507.
[153] Malek A, Momeni-Masuleh S H. A mixed collocation-finite difference method for3D microscopicheat transport problems. J. Comput. Appl. Math.,2008,217:137–147.
[154] Lim H, Kim S, Jr. Douglas J. Numerical methods for viscous and nonviscous wave equations.Appl. Numer. Math.,2007,57:194–212.
[155] Karaa S. Approximate factorization for a viscous wave equation. Computing,2010,89:199–215.
[156] Karaa S. Error estimates for finite element approximates of a viscous wave equation. Numer.Func. Anal. Opt.,2011,32:750–767.
[157] Liao H, Sun Z. Maximum norm error estimates of efficient difference schemes for second-orderwave equations. J. Comput. Appl. Math.,2011,235:2217–2233.