Document Zbl 07927805

Liu, Li-Bin; Ye, Limin; Bao, Xiaobing; Zhang, Yong

A second order numerical method for a Volterra integro-differential equation with a weakly singular kernel. (English) Zbl 07927805

Netw. Heterog. Media 19, No. 2, 740-752 (2024).

Summary: In this paper, a second finite difference method on a graded grid is proposed for a Volterra integro-differential equation with a weakly singular kernel. The proposed scheme is obtained by using the two-step backward differentiation formula (BDF2) to discretize the first derivative term and the first-order interpolation scheme to approximate the integral term. The analysis of stability is proved and used to prove the convergence of our presented numerical method in the discrete maximum norm. Finally, Numerical experiments are given to verify the theoretical results.

MSC:

65Mxx	Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65Lxx	Numerical methods for ordinary differential equations
65Rxx	Numerical methods for integral equations, integral transforms

Keywords:

Volterra integro-differential equation; weakly singular kernel; orthogonal convolution kernels; graded mesh

Cite Review PDF

Full Text: DOI

References:

[1]	H. Brunner, D. Schötzau, hp-Discontinuous galerkin time-stepping for Volterra integro differential equattions, SIAM J Numer Anal, 44 (2006), 224-245. https://doi.org/10.1137/040619314 doi: 10.1137/040619314 · Zbl 1116.65127 · doi:10.1137/040619314
[2]	A. M. Wazwaz, Volterra Integro-Differential Equations, Berlin Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-21449-3_5 · doi:10.1007/978-3-642-21449-3_5
[3]	B. Hermann, Volterra integral equations: an introduction to theory and applications, Cambridge: Cambridge University Press, 2017. https://doi.org/10.1017/9781316162491 · Zbl 1376.45002 · doi:10.1017/9781316162491
[4]	I. Amirali, B. Fedaka, G. M. Amiraliyev, On the second-order neutral Volterra integro-differential equation and its numerical solution, Appl. Math. Comput., 476 (2024), 128765. https://doi.org/10.1016/j.amc.2024.128765 doi: 10.1016/j.amc.2024.128765 · doi:10.1016/j.amc.2024.128765
[5]	H. Chen, W. Qiu, M. A. Zaky, A. S. Hendy, A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel, Calcolo, 60 (2023), 13-42. https://doi.org/10.1007/s10092-023-00508-6 doi: 10.1007/s10092-023-00508-6 · Zbl 1508.65100 · doi:10.1007/s10092-023-00508-6
[6]	I. Amirali, H. Acar, Stability inequalities and numerical solution for neutral Volterra delay integro-differential equation, J. Comput. Appl. Math., 436 (2024), 115343. https://doi.org/10.1016/j.cam.2023.115343 doi: 10.1016/j.cam.2023.115343 · Zbl 1522.65252 · doi:10.1016/j.cam.2023.115343
[7]	M. Kassem, A superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels, Math. Comput., 82 (2013), 1987-2005. http://doi:10.1090/s0025-5718-2013-02689-0 doi: 10.1090/s0025-5718-2013-02689-0 · Zbl 1273.65198 · doi:10.1090/s0025-5718-2013-02689-0
[8]	Z. Wang, Y. Guo, L. Yi, An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels, Math. Comput., 86 (2017), 2285-2324. https://doi.org/10.1090/mcom/3183 doi: 10.1090/mcom/3183 · Zbl 1364.65299 · doi:10.1090/mcom/3183
[9]	Y. X. Wei, Y. P. Chen, Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions, Adv Appl Math Mech, 4 (2012), 1-20. https://doi.org/10.4208/aamm.10-m1055 doi: 10.4208/aamm.10-m1055 · Zbl 1262.45005 · doi:10.4208/aamm.10-m1055
[10]	C. L. Wang, B. Y. Chen, An hp-version spectral collocation method for nonlinear Volterra integro-differential equation with weakly singular kernels, AIMS. Math, 8 (2023), 19816-19841. http://doi.org/10.3934/math.20231010 doi: 10.3934/math.20231010 · doi:10.3934/math.20231010
[11]	H. L. Jia, Y. Yang, Z. Q. Wang, An hp-version Chebyshev spectral collocation method for nonlinear Volterra integro-differential equations with weakly singular kernels, Numer. Math. Theor. Meth. Appl., 12 (2019), 969-994. http://doi.org/10.4208/nmtma.OA-2018-0104 doi: 10.4208/nmtma.OA-2018-0104 · Zbl 1449.65274 · doi:10.4208/nmtma.OA-2018-0104
[12]	X. M. Wang, An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations, Numer. Math., 121 (2012), 753-779. http://doi.org/10.1007/s00211-012-0450-3 doi: 10.1007/s00211-012-0450-3 · Zbl 1435.76054 · doi:10.1007/s00211-012-0450-3
[13]	K. L. Cheng, C. Wang, Long time stability of high order multistep numerical schemes for two-dimensional incompressible Navier-Stokes equations, SIAM. J. Numer. Anal., 54 (2016), 3123-3144. http://doi.org/10.1137/16M1061588 doi: 10.1137/16M1061588 · Zbl 1457.65141 · doi:10.1137/16M1061588
[14]	H. L. Liao, Z. Zhang, Analysis of adaptive BDF2 scheme for diffusion equations, Math. Comp., 90 (2020), 1207-1226. https://doi.org/10.1090/mcom/3585 doi: 10.1090/mcom/3585 · Zbl 1466.65067 · doi:10.1090/mcom/3585
[15]	H. Liao, X. Song, T. Tang, T. Zhou, Analysis of the second order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection, Sci. China Math., 64 (2021), 887-902. https://doi.org/10.1007/s11425-020-1817-4 doi: 10.1007/s11425-020-1817-4 · Zbl 1467.74092 · doi:10.1007/s11425-020-1817-4
[16]	W. S. Wang, M. L. Mao, Z. Wang, Stability and error estimates for the variable step-size BDF2 method for linear and semilinear parabolic equations, Adv. Comput. Math., 47 (2021), 8-35. https://doi.org/10.1007/s10444-020-09839-2 doi: 10.1007/s10444-020-09839-2 · Zbl 1472.65108 · doi:10.1007/s10444-020-09839-2
[17]	W. Wang, Z. Wang, M. Mao, Linearly implicit variable step-size BDF schemes with Fourier pseudospectral approximation for incompressible Navier-Stokes equations, Appl. Numer. Math., 172 (2022), 393-412. https://doi.org/10.1016/j.apnum.2021.10.019 doi: 10.1016/j.apnum.2021.10.019 · Zbl 1484.65276 · doi:10.1016/j.apnum.2021.10.019
[18]	Y. L. Zhao, X. M. Gu, A. A. Ostermann, A preconditioning technique for an all-at-once system from Volterra subdiffusion equations with graded time steps, J Sci Comput, 88 (2021), 11. https://doi.org/10.1007/s10915-021-01527-7 doi: 10.1007/s10915-021-01527-7 · Zbl 1468.76055 · doi:10.1007/s10915-021-01527-7
[19]	C. P. Li, Q. Yi, A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys., 316 (2016), 614-631. https://doi.org/10.1016/j.jcp.2016.04.039 doi: 10.1016/j.jcp.2016.04.039 · Zbl 1349.65246 · doi:10.1016/j.jcp.2016.04.039
[20]	T. Linß, Error expansion for a first-order upwind difference scheme applied to a model convection-diffusion problem, IMA. J. Numer. Anal., 24 (2004), 239-253. http://doi.org/10.1093/imanum/24.2.239 · Zbl 1060.65085 · doi:10.1093/imanum/24.2.239
[21]	H. L. Liao, T. Tang, T. Zhou, Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equation, J. Comput. Math. 41 (2023), 325-344. https://doi.org/10.4208/jcm.2207-m2022-0020 doi: 10.4208/jcm.2207-m2022-0020 · Zbl 1515.65222 · doi:10.4208/jcm.2207-m2022-0020

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.