A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations. Through a special formulation, a system of nonlinear partial differential equations was solved alternately and explicitly in time without linearizing the nonlinearity. Coupled compact schemes of sixth order accuracy in space were developed to obtain numerical solutions. Within couple compact schemes, variables and their first and second derivatives were solved altogether. The sixth order accuracy in space is achieved with a memory-saving arrangement of state variables so that the linear system is banded instead of blocked. This facilitates solving very large systems. The efficiency, simplicity, and accuracy make this coupled compact method viable as variational and weighted residual methods. Results were compared with exact solutions which were obtained via devised forcing terms. Error analyses were carried out, and the sixth order convergence in space and second order convergence in time were demonstrated. Long time integration was also studied to show stability and error convergence rates. 1. Introduction Compact difference methods, essentially the implicit versions of finite different methods, are superior to the explicit versions in achieving high order accuracy. High order compact methods directly approximate all derivatives with high accuracy without any variational operation or projection. They are very effective for complex and strongly nonlinear partial differential equations, especially when variational methods are inconvenient to implement. They have smaller stencils than explicit finite difference methods while maintaining high order accuracy. Much development has been achieved in finite difference methods. A large number of papers related to various finite difference algorithms have been published over the past decades. Kreiss [1] was among the first who pioneered the research on compact implicit methods. Hirsh [2] developed and applied a 4th order compact scheme to Burgers’ equation in fluid mechanics. Adam [3] developed 4th order compact schemes for parabolic equations on uniform grids. Hoffman [4] studied the truncation errors of the centered finite difference method on both uniform and nonuniform grids and discussed the accuracy of these schemes after applying grid transformations. Dennis and Hudson [5] studied a 4th order compact scheme to approximate Navier-Stokes type operators. Rai and Moin [6] presented several finite difference solutions for an incompressible turbulence channel flow. They
References
[1]
H. O. Kreiss, “Methods for the approximate solution of time dependent problems,” GARP Report No. 13, 1975.
[2]
R. S. Hirsh, “Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique,” Journal of Computational Physics, vol. 19, no. 1, pp. 90–109, 1975.
[3]
Y. Adam, “Highly accurate compact implicit methods and boundary conditions,” Journal of Computational Physics, vol. 24, no. 1, pp. 10–22, 1977.
[4]
J. D. Hoffman, “Relationship between the truncation errors of centered finite-difference approximations on uniform and nonuniform meshes,” Journal of Computational Physics, vol. 46, no. 3, pp. 469–474, 1982.
[5]
S. C. R. Dennis and J. D. Hudson, “Compact h4 finite-difference approximations to operators of Navier-Stokes type,” Journal of Computational Physics, vol. 85, no. 2, pp. 390–416, 1989.
[6]
M. M. Rai and P. Moin, “Direct simulations of turbulent flow using finite-difference schemes,” Journal of Computational Physics, vol. 96, no. 1, pp. 15–53, 1991.
[7]
S. K. Lele, “Compact finite difference schemes with spectral-like resolution,” Journal of Computational Physics, vol. 103, no. 1, pp. 16–42, 1992.
[8]
S.-T. Yu, L. S. Hultgren, and N.-S. Liu, “Direct calculations of waves in fluid flows using high-order compact difference scheme,” AIAA Journal, vol. 32, no. 9, pp. 1766–1773, 1994.
[9]
K. Mahesh, “A family of high order finite difference schemes with good spectral resolution,” Journal of Computational Physics, vol. 145, no. 1, pp. 332–358, 1998.
[10]
W. Dai and R. Nassar, “A compact finite difference scheme for solving a three-dimensional heat transport equation in a thin film,” Numerical Methods for Partial Differential Equations, vol. 16, no. 5, pp. 441–458, 2000.
[11]
W. Dai and R. Nassar, “A compact finite-difference scheme for solving a one-dimensional heat transport equation at the microscale,” Journal of Computational and Applied Mathematics, vol. 132, no. 2, pp. 431–441, 2001.
[12]
H. Sun and J. Zhang, “A high-order finite difference discretization strategy based on extrapolation for convection diffusion equations,” Numerical Methods for Partial Differential Equations, vol. 20, no. 1, pp. 18–32, 2004.
[13]
J. A. Ekaterinaris, “Implicit, high-resolution, compact schemes for gas dynamics and aeroacoustics,” Journal of Computational Physics, vol. 156, no. 2, pp. 272–299, 1999.
[14]
M. Li, T. Tang, and B. Fornberg, “A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations,” International Journal for Numerical Methods in Fluids, vol. 20, no. 10, pp. 1137–1151, 1995.
[15]
J. S. Shang, “High-order compact-difference schemes for time-dependent Maxwell equations,” Journal of Computational Physics, vol. 153, no. 2, pp. 312–333, 1999.
[16]
W. F. Spotz and G. F. Carey, “Extension of high-order compact schemes to time-dependent problems,” Numerical Methods for Partial Differential Equations, vol. 17, no. 6, pp. 657–672, 2001.
[17]
H. El-Zoheiry, “Numerical study of the improved Boussinesq equation,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 377–384, 2002.
[18]
J. Zhang, H. Sun, and J. J. Zhao, “High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems,” Journal of Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 41-42, pp. 4661–4674, 2002.
[19]
T. K. Sengupta, G. Ganeriwal, and S. De, “Analysis of central and upwind compact schemes,” Journal of Computational Physics, vol. 192, no. 2, pp. 677–694, 2003.
[20]
M. Y. Shen, Z. B. Zhang, and X. L. Niu, “A new way for constructing high accuracy shock-capturing generalized compact difference schemes,” Journal of Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 25, pp. 2703–2725, 2003.
[21]
A. Jocksch, N. A. Adams, and L. Kleiser, “An asymptotically stable compact upwind-biased finite-difference scheme for hyperbolic systems,” Journal of Computational Physics, vol. 208, no. 2, pp. 435–454, 2005.
[22]
M. A. H. Mohamad, S. Basri, I. Rahim, and M. Mohamad, “An AUSM-based third-order compact scheme for solving Euler equations,” International Journal of Engineering & Technology, vol. 3, no. 1, pp. 1–12, 2006.
[23]
S. L. Anthonio and K. R. Hall, “High-order compact numerical schemes for non-hydrostatic free surface flows,” International Journal for Numerical Methods in Fluids, vol. 52, no. 12, pp. 1315–1337, 2006.
[24]
E. C. Cruz and Q. Chen, “Numerical modeling of nonlinear water waves over heterogeneous porous beds,” Ocean Engineering, vol. 34, no. 8-9, pp. 1303–1321, 2007.
[25]
P. H. Chiu and T. W. H. Sheu, “On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection-diffusion equation,” Journal of Computational Physics, vol. 228, no. 10, pp. 3640–3655, 2009.
[26]
Q. Zhou, Z. Yao, F. He, and M. Y. Shen, “A new family of high-order compact upwind difference schemes with good spectral resolution,” Journal of Computational Physics, vol. 227, no. 2, pp. 1306–1339, 2007.
[27]
Y. V. S. S. Sanyasiraju and N. Mishra, “Spectral resolutioned exponential compact higher order scheme (SRECHOS) for convection-diffusion equations,” Journal of Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 51-52, pp. 4737–4744, 2008.
[28]
M. Dehghan and A. Mohebbi, “High-order compact boundary value method for the solution of unsteady convection-diffusion problems,” Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 683–699, 2008.
[29]
A. Shah, H. Guo, and L. Yuan, “A third-order upwind compact scheme on curvilinear meshes for the incompressible Navier-Stokes equations,” Communications in Computational Physics, vol. 5, no. 2-4, pp. 712–729, 2009.
[30]
S.-S. Xie, G.-X. Li, and S. Yi, “Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schr？dinger equation,” Journal of Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 9-12, pp. 1052–1060, 2009.
[31]
M. Tonelli and M. Petti, “Hybrid finite volume—finite difference scheme for 2DH improved Boussinesq equations,” Coastal Engineering, vol. 56, no. 5-6, pp. 609–620, 2009.
[32]
R. L. Burden and J. D. Faires, Numerical Analysis, Thomson Brooks/Cole, Belmont, Calif, USA, 8th edition, 2005.
[33]
D. Liu, W. Kuang, and A. Tangborn, “High-order compact implicit difference methods for parabolic equations in geodynamo simulation,” Advances in Mathematical Physics, vol. 2009, Article ID 568296, 23 pages, 2009.
[34]
L. Gamet, F. Ducros, F. Nicoud, and T. Poinsot, “Compact finite difference schemes on nonuniform meshes: application to direct numerical simulations of compressible flows,” International Journal for Numerical Methods in Fluids, vol. 29, pp. 159–191, 1999.
[35]
O. V. Vasilyev, “High order finite difference schemes on non-uniform meshes with good conservation properties,” Journal of Computational Physics, vol. 157, pp. 746–761, 2000.
[36]
M. Hermanns and J. A. Hernández, “Stable high-order finite-difference methods based on non-uniform grid point distributions,” International Journal for Numerical Methods in Fluids, vol. 56, no. 3, pp. 233–255, 2008.
[37]
M. W. Dingemans, Water Wave Propagation over Uneven Bottoms, World Scientific, 1997.
[38]
B. Gustafsson, H. Kreiss, and J. Oliger, Time Dependent Problems and Difference Methods, John Wiley & Sons, 1995.
[39]
D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Society for Industrial and Applied Mathematics, 1977.
[40]
C. Pozrikidis, Introduction to Finite and Spectal Element Method Using MATLAB, Chapman & Hall/CRC, 2005.
