A second-order box solver for nonlinear delayed convection-diffusion equations with Neumann boundary conditions

ABSTRACT

In this paper, by applying order reduction approach, a second-order accurate box scheme is established to solve a nonlinear delayed convection-diffusion equations with Neumann boundary conditions. By the discrete energy method, it is shown that the difference scheme is uniquely solvable, and has a convergence rate of $\mathcal{O}(\mathrm{\Delta }{t}^2+h^2)$ with respect to $L^2$- norm in constrained and non-constrained temporal grids. Besides, for constrained temporal step, a Richardson extrapolation method (REM) used along with the box scheme, which makes final solution third-order accurate in both time and space, is developed in detail. Finally, numerical results confirm the accuracy and efficiency of our solvers.

KEYWORDS:

• Nonlinear convection-diffusion equations with delays
• Neumann boundary conditions
• box scheme
• convergence
• solvability

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M15

Acknowledgments

Authors are very grateful to Principal Editor Professor Choi-Hong Lai, and referees for their valuable comments and suggestions, which have greatly improved the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Dingwen Deng's work was partially supported by National Natural Science Foundation of China (NSFC) (Grant Nos. 11861047), NSF of Jiangxi Provincial Education Department (Grant Nos. GJJ160706, DA201807160), Postdoctoral NSF of China (Grant No. 2015M582631), Postdoctoral NSF of Shannxi province (Grant No. 2016BSHYDZZ35), State Scholarship Fund of CSC for Overseas Studies (Grant No. 201608360086). He also thanks Department of Mathematics and Statistics, York university for his visit. Yaolin Jiang's work is supported partially by NSFC (No. 11871393). Dong Liang's work is supported partially by Natural Sciences and Engineering Research Council of Canada.