An ADI finite difference method for the two-dimensional Volterra integro-differential equation with weakly singular kernel

ABSTRACT

In this paper, we study the two-dimensional Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain. The memory kernel of this equation makes it not easy to construct an efficient numerical scheme and perform theoretical analysis. The numerical method is considered by the finite difference approach for spatial discretization and Crank–Nicolson (CN) alternating direction implicit (ADI) scheme in the time direction. The integral terms are approximated by the fractional convolution quadrature. We prove that the proposed method is unconditional stable and derive error estimates in $L^2$ norm. The literature reported on the ADI finite difference method of this model is extremely sparse. Numerical results support the theoretical analysis.

KEYWORDS:

• Volterra integro-differential equation
• ADI finite difference scheme
• convolution quadrature
• stability
• convergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M15

Acknowledgments

The authors would like to thank the editor and reviewers for their constructive comments and suggestions, which helped the authors to improve the quality of the paper significantly.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was partly supported by the National Natural Science Foundation of China [No. 12101080, 12126308, 11701103], Young Top-notch Talent Program of Guangdong Province [No. 2017GC010379], Natural Science Foundation of Guangdong Province [No. 2022A1515012147, 2019A1515010876], the Project of Science and Technology of Guangzhou [No. 201904010341, 202102020704], and the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University [2021023], Scientific Research Fund of Hunan Provincial Education Department [No. 21C0188].