A new coupled high-order compact method for the three-dimensional nonlinear biharmonic equations

Abstract

This paper presents a new family of fourth- compact finite difference schemes for the numerical solution of three-dimensional nonlinear biharmonic equations using coupled approach. The numerical solutions of unknown variable and its first- derivatives as well as v(=Δ u) are obtained not only in the interior but also at the boundary. A prominent contribution of this work is that the boundary conditions for the variable v are approximated more accurately, which plays an important role for the efficiency of calculation. Finally, numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these schemes, including the steady Navier–Stokes equation in terms of vorticity-stream function formulation.

Keywords:

• 3D nonlinear biharmonic equation
• Dirichlet boundary conditions of first kind
• compact difference scheme
• different composition
• high- numerical method

2000 AMS Subject Classifications:

• 65M60
• 65M15
• 35K55

Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this paper. The authors also thank Dr. Dongwei Gui (Cele National Station of Observation & Research for Desert-Grassland Ecosystem in Xinjiang) for his support and encouragement in this work. The first author is partially supported by the Graduate Student Research Innovation Program of Xinjiang (No. XJGRI2013011) and the Excellent Doctor Innovation Program of Xinjiang University (No. XJUBSCX-2012003). The second author is partially supported by the Distinguished Young Scholars Fund of Xinjiang Province (No. 201311010), the NSF of China (No. 11271313, No. 61163027) and the Key Project of Chinese Ministry of Education (No. 212197).