Decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with smooth initial data

ABSTRACT

Based on the mixed finite element method, we consider the decoupled Crank–Nicolson/Adams–Bashforth scheme for the Boussinesq equations with smooth initial data in this paper. The temporal treatment of the spatial discrete Boussinesq equations is based on the implicit Crank–Nicolson scheme for the linear terms and the explicit Adams–Bashforth scheme for the nonlinear terms. Thanks to the decoupled method, the considered problem is split into two subproblems and these subproblems can be solved in parallel. Under some restriction on the time step, we present the stability and convergence results of numerical solutions, Finally, some numerical experiments are provided to test the performance of the developed numerical scheme and verify the established theoretical findings.

Keywords:

• The Boussinesq equations
• Crank–Nicolson/Adams–Bashforth scheme
• decoupled method
• stability
• convergence

2000 MSC Classifications:

• 65N15
• 65N30
• 76D07

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The Foundation for University Key Teacher by the Henan Province (2016GGJS-045), and the Foundation of Distinguished Young Scientists of HPU (J2015-05) and the NSF of China (No. 11701153).