Third-order temporal discrete scheme for the non-stationary Navier–Stokes equations

Abstract

In this article, we suggest a new third-order time discrete scheme for the two-dimensional non-stationary Navier–Stokes equations. After presenting the Galerkin finite element approximation for the spatial discretization, we consider an implicit/explicit time discrete scheme for the problem, which is based on the two-step Adams–Moulton scheme (implicit scheme) for the linear term and the three-step Adams–Bashforth scheme (explicit scheme) for the nonlinear term. In this method, we only need to solve a linearized discrete system at each time step, so the scheme can converge fast and the computational cost can be reduced. Moreover, under some assumptions, we deduce the stability and optimal error estimate for the velocity in L ²-norm.

Keywords:

• Navier–Stokes equations
• time discretization
• third-order scheme
• Adams–Bashforth scheme
• error estimate

2000 AMS Subject Classifications::

• 35L70
• 65M30
• 76D06

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 11071193). The authors thank the editor and reviewers for their valuable comments and suggestions which helped to improve the results of this paper.