A new high accuracy method for two-dimensional biharmonic equation with nonlinear third derivative terms: application to Navier–Stokes equations of motion

References

• Y. Adam, Highly accurate compact implicit methods and boundary conditions, J. Comput. Phys. 24 (1977), pp. 10–22. doi: 10.1016/0021-9991(77)90106-1
   Web of Science ®Google Scholar
• L. Bauer and E.L. Riess, Block five diagonal matrices and the fast numerical solution of the biharmonic equation, Math. Comput. 26 (1972), pp. 311–326. doi: 10.1090/S0025-5718-1972-0312751-9
   Web of Science ®Google Scholar
• R.F. Boisvert, Families of high order accurate discretizations of some elliptic equations, SIAM J. Sci. Statist. Comput. 2 (1981), pp. 268–284.
   Web of Science ®Google Scholar
• C.H. Bruneau and C. Jouron, An efficient scheme for solving steady incompressible Navier–Stokes equations, J. Comput. Phys. 89 (1990), 389–413. doi: 10.1016/0021-9991(90)90149-U
   Web of Science ®Google Scholar
• W. Dai, An improved compact finite difference scheme for solving an N-carrier system with Neumann boundary conditions, Numer. Meth. Partial Differential Equations 27 (2011), pp. 436–446. doi: 10.1002/num.20531
   Web of Science ®Google Scholar
• W. Dai and D.Y. Tzou, A fourth order compact finite difference scheme for solving an N-carrier system with Neumann boundary conditions, Numer. Meth. Partial Differential Equations 26 (2010), pp. 274–289.
   Google Scholar
• M. Dehghan and A. Mohebbi, Solution of the two dimensional second biharmonic equation with high-order accuracy, Kybernetes 37 (2008), pp. 1165–1179. doi: 10.1108/03684920810884964
   Web of Science ®Google Scholar
• L.W. Ehrlich, Solving the biharmonic equation as coupled finite difference equations, SIAM. J. Numer. Anal. 8 (1971), pp. 278–287. doi: 10.1137/0708029
   Web of Science ®Google Scholar
• L.W. Ehrlich, Point and block SOR applied to a coupled set of difference equations, Computing 12 (1974), pp. 181–194. doi: 10.1007/BF02293104
   Web of Science ®Google Scholar
• D.J. Evans and R.K. Mohanty, Block iterative methods for the numerical solution of two dimensional nonlinear biharmonic equations, Int. J. Comput. Math. 69 (1998), pp. 371–390. doi: 10.1080/00207169808804729
   Web of Science ®Google Scholar
• U. Ghia, K.N. Ghia, and C.T. Shin, High resolution for incompressible Navier–Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982), pp. 387–411. doi: 10.1016/0021-9991(82)90058-4
   Web of Science ®Google Scholar
• L.A. Hageman and D.M. Young, Applied Iterative Methods, Dover Publication, New York, 2012.
   Google Scholar
• R.S. Hirsh, Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys. 19 (1975), pp. 90–109. doi: 10.1016/0021-9991(75)90118-7
   Web of Science ®Google Scholar
• M.K. Jain, R.K. Jain, and R.K. Mohanty, Fourth order difference methods for the system of 2D nonlinear elliptic partial differential equations, Numer. Meth. Partial Differential Equations 7 (1991), pp. 227–244. doi: 10.1002/num.1690070303
   Google Scholar
• C.T. Kelly, Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, PA, 1995.
   Google Scholar
• Y. Kwon, R. Manohar, and J.W. Stephenson, Single cell fourth order methods for the biharmonic equation, Congr. Numer. 34 (1982), pp. 475–482.
   Google Scholar
• S.K. Lele, Compact finite difference schemes with spectral like resolution, J. Comput. Phys. 103 (1992), pp. 16–42. doi: 10.1016/0021-9991(92)90324-R
   Web of Science ®Google Scholar
• M. Li, T. Tang, and B. Fornberg, A compact fourth order finite difference scheme for the steady incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids 20 (1995), pp. 1137–1151. doi: 10.1002/fld.1650201003
   Web of Science ®Google Scholar
• R.K. Mohanty, Order h4 difference methods for a class of singular two space elliptic boundary value problems, J. Comput. Appl. Math. 81 (1997), pp. 229–247. doi: 10.1016/S0377-0427(97)00058-7
   Web of Science ®Google Scholar
• R.K. Mohanty, A new high accuracy finite difference discretization for the solution of 2D nonlinear biharmonic equations using coupled approach, Numer. Meth. Partial Differential Equations 26 (2010), pp. 931–944.
   Web of Science ®Google Scholar
• R.K. Mohanty and S. Dey, A new finite difference discretization of order four for (∂u/∂n) for two dimensional quasilinear elliptic boundary value problem, Int. J. Comput. Math. 76 (2001), pp. 505–516. doi: 10.1080/00207160108805043
   Web of Science ®Google Scholar
• R.K. Mohanty, M.K. Jain, and P.K. Pandey, Finite difference methods of order two and four for 2-D nonlinear biharmonic problems of first kind, Int. J. Comput. Math. 61 (1996), pp. 155–163. doi: 10.1080/00207169608804507
   Web of Science ®Google Scholar
• L. Mu, J. Wang, and X. Ye, Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, Numer. Meth. Partial Differential Equations 30 (2014), pp. 1003–1029. doi: 10.1002/num.21855
   Web of Science ®Google Scholar
• S.V. Parter, Block iterative methods, in Elliptic Problem Solvers, M.H. Schultz, ed., Academic Press, New York, 1981, pp. 247–253.
   Google Scholar
• Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, PA, 2003.
   Google Scholar
• J. Smith, The coupled equation approach to the numerical solution of the biharmonic equation by finite differences, SIAM. J. Numer. Anal. 5 (1970), pp. 104–111. doi: 10.1137/0707005
   Web of Science ®Google Scholar
• W.F. Sportz and G.F. Carey, High order compact scheme for the steady stream-function vorticity equation, Int. J. Numer. Meth. Eng. 38 (1995), pp. 3497–3512. doi: 10.1002/nme.1620382008
   Web of Science ®Google Scholar
• J.W. Stephenson, Single cell discretizations of order two and four for biharmonic problems, J. Comput. Phys. 55 (1984), pp. 65–80. doi: 10.1016/0021-9991(84)90015-9
   Web of Science ®Google Scholar
• Z.F. Tian and S.Q. Dai, High order compact exponential finite difference methods for convection-diffusion type problems, J. Comput. Phys. 220 (2007), pp. 952–974. doi: 10.1016/j.jcp.2006.06.001
   Web of Science ®Google Scholar
• I. Yavneh, Analysis of a fourth order compact scheme for convection–diffusion, J. Comput. Phys. 133 (1997), pp. 361–364. doi: 10.1006/jcph.1997.5659
   Web of Science ®Google Scholar