%0 Journal Article
%T A Sixth Order Accuracy Solution to a System of Nonlinear Differential Equations with Coupled Compact Method
%A Don Liu
%A Qin Chen
%A Yifan Wang
%J Journal of Computational Engineering
%D 2013
%R 10.1155/2013/432192
%X 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
%U http://www.hindawi.com/journals/jcengi/2013/432192/