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Lu, Jiashu; Zhang, Qingyu; Zhao, Lijing; Nie, Yufeng

An efficient Jacobi spectral collocation method with nonlocal quadrature rules for multi-dimensional volume-constrained nonlocal models. (English) Zbl 07714972

Int. J. Comput. Methods 20, No. 5, Article ID 2350004, 16 p. (2023).
Summary: In this paper, an efficient Jacobi spectral collocation method is developed for multi-dimensional weakly singular volume-constrained nonlocal models including both nonlocal diffusion (ND) models and peridynamic (PD) models. The model equation contains a weakly singular integral operator with the singularity located at the center of the integral domain, and the numerical approximation of it becomes an essential difficulty in solving nonlocal models. To approximate such singular nonlocal integrals in an accurate way, a novel nonlocal quadrature rule is constructed to accurately compute these integrals for the numerical solutions produced by spectral methods. Numerical experiments are given to show that spectral accuracy can be obtained by using the proposed Jacobi spectral collocation methods for several different nonlocal models. Besides, we numerically verify that the numerical solution of our Jacobi spectral method can converge to its correct local limit as the nonlocal interactions vanish.

MSC:

65-XX Numerical analysis
81-XX Quantum theory

Keywords:

nonlocal models; nonlocal quadrature rules; spectral collocation methods; weakly singular kernel
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References:

[1] Canuto, C., Hussaini, N. Y., Quarteroni, A. and Zang, T. A. [2007] Spectral Methods: Fundamentals in Single Domains, 1st Edition (Springer Science and Business Media, Germany). · Zbl 1093.76002
[2] Das, P. and Rana, S. [2021] “ Theoretical prospects of fractional order weakly singular Volterra integro differential equations and their approximations with convergence analysis,” Math. Methods Appl. Sci.44, 9419-9440. · Zbl 1512.65132
[3] Das, P., Rana, S. and Ramos, H. [2022] “ On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis,” J. Comput. Appl. Math.404, 113116. · Zbl 1481.65265
[4] Dayal, K. and Bhattacharya, K. [2020] “ Kinetics of phase transformations in the peridynamic formulation of continuum mechanics,” J. Mech. Phys. Solids54, 1811-1842. · Zbl 1120.74690
[5] Du, Q., Gunzburger, M., Lehoucq, R. B. and Zhou, K. [2012] “ Analysis and approximation of nonlocal diffusion problems with volume constraints,” SIAM Rev.54, 667-696. · Zbl 1422.76168
[6] Du, Q., Huang, Z. and Lehoucq, R. B. [2014] “ Nonlocal convection-diffusion volume-constrained problems and jump processes,” Discrete Contin. Dyn. Syst. B19, 373-389. · Zbl 1284.35223
[7] Du, N., Guo, X. and Wang, H. [2020] “ A fast state-based peridynamic numerical model,” Commun. Comput. Phys.27, 274-291. · Zbl 1473.65037
[8] Du, Q., Tian, X. C. and Zhou, Z. [2022] “Nonlocal diffusion models with consistent local and fractional limits,” https://arXiv.org/abs/2203.00167.
[9] Ghoreishi, F. and Mokhtary, P. [2014] “ Spectral collocation method for multi-order fractional differential equations,” Int. J. Comput. Methods11, 1350072. · Zbl 1359.65213
[10] Huang, K. and Du, Q. [2022] “ Stability of a nonlocal traffic flow model for connected vehicles,” SIAM J. Appl. Math.82, 221-243. · Zbl 1486.35293
[11] Kharazmi, E. and Zayernouri, M. [2018] “ Fractional pseudo-spectral methods for distributed-order fractional PDEs,” Int. J. Comput. Math.95, 1340-1361. · Zbl 1513.65251
[12] Kumar, S., Nieto, J. J. and Ahmad, B. [2022] “ Chebyshev spectral method for solving fuzzy fractional Fredholm-Volterra integro-differential equation,” Math. Comput. Simul.192, 501-513. · Zbl 1530.65186
[13] Lu, J. S. and Nie, Y. F. [2021] “ A collocation method based on localized radial basis functions with reproducibility for nonlocal diffusion models,” Comput. Appl. Math.40, 271. · Zbl 1476.65343
[14] Lu, J. S. and Nie, Y. F. [2022] “ A reduced-order fast reproducing kernel collocation method for nonlocal models with inhomogeneous volume constraints,” Comput. Math. Appl.121, 52-61. · Zbl 1524.65669
[15] Lu, J. S., Yang, M. N. and Nie, Y. F. [2022] “ Convergence analysis of Jacobi spectral collocation methods for weakly singular nonlocal diffusion equations-with volume constraints,” Appl. Math. Comput.431, 127345. · Zbl 1510.65262
[16] Pasetto, M., Leng, Y., Chen, J. S., Foster, J. T. and Seleson, P. [2018] “ Reproducing kernel enhanced approach for peridynamic solutions,” Comput. Methods Appl. Mech. Eng.340, 1044-1078. · Zbl 1440.74478
[17] Pasetto, M., Shen, Z. X., Delia, M., Tian, X. C., Trask, N. and Kamensky, D. [2022] “ Efficient optimization-based quadrature for variational discretization of nonlocal problems,” Comput. Methods Appl. Mech. Eng.396, 115104. · Zbl 1507.74502
[18] Seleson, P. and Littlewood, D. J. [2016] “ Convergence studies in meshfree peridynamic simulation, computers & mathematics with applications,” Comput. Methods Appl. Mech. Eng.71, 2432-2448. · Zbl 1443.65253
[19] Shen, J., Tang, T. and Wang, L. L. [2011] Spectral Methods: Algorithms, Analysis and Applications, 1st Edition (Springer Science and Business Media, New York). · Zbl 1227.65117
[20] Silling, S. A., Epton, M., Weckner, O., Xu, J. F. and Askari, E. [2007] “ Peridynamic states and constitutive modeling,” J. Elast.88, 151-184. · Zbl 1120.74003
[21] Silling, S. A. [2000] “ Reformulation of elasticity theory for discontinuities and long-range forces,” J. Mech. Phys. Solids48, 175-209. · Zbl 0970.74030
[22] Tian, X. C. and Engquist, B. [2019] “ Fast algorithm for computing nonlocal operators with finite interaction distance,” Comput. Methods Appl. Mech. Eng.17, 1653-1670. · Zbl 1428.82083
[23] Tian, H., Zhang, J. and Ju, L. L. [2020] “ A spectral collocation method for nonlocal diffusion equations with volume constrained boundary conditions,” Appl. Math. Comput.370, 124930. · Zbl 1433.65321
[24] Trask, N., You, H. Q., Yu, Y. and Parks, M. L. [2019] “ An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics,” Comput. Methods Appl. Mech. Eng.343, 151-165. · Zbl 1440.74463
[25] Wang, H. and Tian, H. [2014] “ A fast and faithful collocation method with efficient matrix assembly for a two-dimensional nonlocal diffusion model,” Comput. Methods Appl. Mech. Eng.273, 19-36. · Zbl 1296.65172
[26] Witman, D. R., Gunzburger, M. and Peterson, J. [2017] “ Reduced-order modeling for nonlocal diffusion problems,” Int. J. Numer. Methods Fluids83, 307-327.
[27] You, H. Q., Yu, Y., D’Elia, M., Gao, T. and Silling, S. A. [2022] “Nonlocal Kernel Network (NKN): A stable and resolution-independent deep neural network,” https://arXiv.org/abs/2201.02217. · Zbl 07592139
[28] Zhang, S. Y. and Nie, Y. F. [2020] “ A POD-based fast algorithm for the nonlocal unsteady problems,” Int. J. Numer. Anal. Model.17, 858-871. · Zbl 1482.65190
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