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

Abstract

In this paper, we propose a new compact fourth-order accurate method for solving the two-dimensional fourth-order elliptic boundary value problem with third-order nonlinear derivative terms. We use only 9-point single computational cell in the scheme. The proposed method is then employed to solve Navier–Stokes equations of motion in terms of streamfunction–velocity formulation, and the lid-driven square cavity problem. We describe the derivation of the method in details and also discuss how our streamfunction–velocity formulation is able to handle boundary conditions in terms of normal derivatives. Numerical results show that the proposed method enables us to obtain oscillation-free high accuracy solution.

Keywords:

• nonlinear biharmonic equation
• normal derivatives
• Navier–Stokes equations of motion
• streamfunction–velocity formulation
• lid-driven square cavity
• block-tridiagonal matrices
• Reynolds number

2010 AMS Subject Classifications:

• 65N06
• 65N12

Acknowledgements

This research was supported by ‘United States-India Educational Foundation’ under ‘2013 Fulbright-Nehru Senior Research Fellowship’ Program, and partially supported by the National Science Foundation (NSF EPS – 1003897).

The authors thank the reviewers for their valuable suggestions, which substantially improved the standard of the paper.