The BDF orthogonal spline collocation method for the two-dimensional evolution equation with memory

ABSTRACT

In this paper, a second-order backward differentiation formula (BDF) orthogonal spline collocation (OSC) method with the truncation error of order $\frac{3}{2}$ in time and $h^{r+1}$ in space is presented for a kind of partial differential equation (PDE) with memory. The stability and convergence of the BDF OSC method in a new norm are proved. Besides, we provide three numerical experiments to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the BDF OSC method. Numerical experiments also exhibit the optimal error estimates in the $L^2$, $L^{\mathrm{\infty }}$ and $H^1$-norms.

KEYWORDS:

• Evolution equation with memory
• second-order BDF
• orthogonal spline collocation
• stability
• convergence

2010 MSC SUBJECT CLASSIFICATIONS:

• 65M70
• 65M12
• 65M15
• 35R11

Acknowledgements

The authors thank the anonymous reviewers for their invaluable comments and suggestions, and thank Prof. Graeme Fairweather for stimulating discussions and for his constant encouragement and support.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [grant numbers 11601144, 11626096]; the Scientific Research Fund of Hunan Provincial Education Department [grant numbers 16K026, 15C0388, 15C0390, YB2016B033]; and the China Postdoctoral Science Foundation [grant number 2016M600964].