Abstract
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.
Highlights
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
The actual accuracy is determined by the effect of the transformation. Another option for nonuniform grids is to use full inclusion of metrics (FIM), such as Liu et al [33] where fourth and sixth order compact methods were used for parabolic partial differential equations encountered in geophysical modeling
Summary
Essentially the implicit versions of finite different methods, are superior to the explicit versions in achieving high order accuracy. Gamet et al [34] developed a 4th order compact scheme for approximating the first order derivative with a full inclusion of metrics directly on a nonuniform mesh with variable grid spacing Using this method, a direct numerical simulation of a compressible turbulent channel flow was conducted. The actual accuracy is determined by the effect of the transformation Another option for nonuniform grids is to use full inclusion of metrics (FIM), such as Liu et al [33] where fourth and sixth order compact methods were used for parabolic partial differential equations encountered in geophysical modeling. One objective of this paper is to develop 6th order coupled compact schemes for modified Boussinesq equations with analytical exact solution for comparison These schemes demonstrate high order accuracy and fast convergence rates in both space and time. Conclusion summarises the novelty and uniqueness of the schemes developed in this paper, and future work is planned
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
