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Fast and high-order accuracy numerical methods for time-dependent nonlocal problems in \(\mathbb{R}^2\). (English) Zbl 1445.74054
Summary: In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of \(\mathcal{O} (\tau^2+h^{4-\gamma})\) with \(0<\gamma <1\), where \(\tau\) and \(h\) are the discretization sizes in the temporal and spatial dimensions, respectively. Also we develop the conjugate gradient squared method for solving the resulting discretized nonsymmetric and indefinite systems arising from time-dependent nonlocal problems including two-dimensional cases. By using additive and multiplicative Cauchy kernels in nonlocal problems, structured coefficient matrix-vector multiplication can be performed efficiently in the conjugate gradient squared iteration. Numerical examples are given to illustrate our theoretical results and demonstrate that the computational cost of the proposed method is of \(O(M \log M)\) operations, where \(M\) is the number of collocation points.
MSC:
74S99 Numerical and other methods in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Software:
CGS
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[1] Andreu-Vaillo, F.; Mazón, JM; Rossi, JD; Toledo-Melero, JJ, Nonlocal Diffusion Problems (2010), Providence: AMS, Providence
[2] Atkinson, KE, The numerical solution of Fredholm integral equations of the second kind, SIAM J. Numer. Anal., 4, 337-348 (1967) · Zbl 0155.47404
[3] Atkinson, KE, The Numerical Solution of Integral Equations of the Second Kind (2009), Cambridge: Cambridge University Press, Cambridge
[4] Bates, P.; Brunner, H.; Zhao, X.; Zou, X., On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations. Fields Institute Communications, 13-52 (2006), Providence: AMS, Providence · Zbl 1101.35073
[5] Chan, RHF; Jin, XQ, An Introduction to Interative Toeplitz Solvers (2007), Phildelphia: SIAM, Phildelphia
[6] Chan, RH; Ng, MK, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38, 427-482 (1996) · Zbl 0863.65013
[7] Chen, MH; Deng, WH, Fourth order accurate scheme for the space fractional diffusion equations, SIAM J. Numer. Anal., 52, 1418-1438 (2014) · Zbl 1318.65048
[8] Chen, MH; Deng, WH, Discretized fractional substantial calculus, ESAIM: Math. Mod. Numer. Anal., 49, 373-394 (2015) · Zbl 1330.65170
[9] Chen, M.H., Ekström, S.E., Serra-Capizzano, S.: A Multigrid method for nonlocal problems: non-diagonally dominant or Toeplitz-plus-tridiagonal systems. SIAM J. Matrix Anal. Appl. (major revised) arXiv:1808.09595
[10] Chen, MH; Deng, WH, Convergence analysis of a multigrid method for a nonlocal model, SIAM J. Matrix Anal. Appl., 38, 869-890 (2017) · Zbl 1373.65068
[11] Chen, M.H., Qi, W.Y., Shi, J.K., Wu, J.M.: A sharp error estimation of piecewise polynomial collocation method for nonlocal problems with weakly singular kernels. IMA J. Numer. Anal. (major revised) arXiv:1909.10756
[12] Chen, MH; Wang, YT; Cheng, X.; Deng, WH, Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation, BIT, 54, 623-647 (2014) · Zbl 1301.35200
[13] Du, Q.; Han, HD; Zhang, JW; Zheng, CX, Numerical solution of a two-dimensional nonlocal wave equation on unbounded domains, SIAM J. Sci. Comput., 40, A1430-A1445 (2018) · Zbl 1392.82036
[14] Du, Q.; Gunzburger, M.; Lehoucq, R.; Zhou, K., Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 56, 676-696 (2012) · Zbl 1422.76168
[15] Du, N.; Wang, H., A fast finite element method for space-fractional dispersion equations on bounded domains in \({\mathbb{R}}^2\), SIAM J. Sci. Comput., 37, A1614-A1635 (2015) · Zbl 1331.65175
[16] de Hoog, F.; Weiss, R., Asymptotic expansions for product integration, Math. Comput., 27, 295-306 (1973) · Zbl 0303.65023
[17] Gao, Y.; Feng, H.; Tian, H.; Ju, LL; Zhang, XP, Nodal-type Newton-Cotes rules for fractional hypersingular integrals, E. Asian J. Appl. Math., 8, 697-714 (2018)
[18] Li, BY; Sun, WW, Newton-Cotes rules for Hadamard finite-part integrals on an interval, IMA J. Numer. Anal., 30, 1235-1255 (2010) · Zbl 1203.65058
[19] Pan, JY; Ng, M.; Wang, H., Fast iterative solvers for linear systems arising from time-dependent space-fractional diffusion equations, SIAM J. Sci. Comput., 38, A2806-A2826 (2016) · Zbl 1348.65067
[20] Pang, H.; Sun, H., Multigrid method for fractional diffusion equations, J. Comput. Phys., 231, 693-703 (2012) · Zbl 1243.65117
[21] Quarteroni, A.; Sacco, R.; Saleri, F., Numerical Mathematics (2007), Berlin: Springer, Berlin · Zbl 0913.65002
[22] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), Philadelphia: SIAM, Philadelphia · Zbl 1002.65042
[23] Silling, SA, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48, 175-209 (2000) · Zbl 0970.74030
[24] Sonneveled, P., CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 10, 36-52 (1989) · Zbl 0666.65029
[25] Tian, H.; Wang, H.; Wang, WQ, An efficient collocation method for a non-local diffusion model, Int. J. Numer. Anal. Model., 4, 815-825 (2013) · Zbl 1280.65134
[26] Varga, RS, Matrix Iterative Analysis (2000), Berlin: Springer, Berlin
[27] Varah, JM, A lower bound for the smallest singular value of a matrix, Linear Algebra Appl., 11, 3-5 (1975) · Zbl 0312.65028
[28] Wang, H.; Basu, T., A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34, A2444-A2458 (2012) · Zbl 1256.35194
[29] Wang, H.; Tian, H., A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model, J. Comput. Phys., 231, 7730-7738 (2012) · Zbl 1254.74112
[30] Wu, JM; Lü, Y., A superconvergence result for the second-order Newton-Cotes formula for certain finite-part integrals, IMA J. Numer. Anal., 25, 253-263 (2005) · Zbl 1069.41026
[31] Zhang, XP; Wu, JM; Ju, LL, An accurate and asymptotically compatible collocation scheme for nonlocal diffusion problems, Appl. Numer. Math., 133, 52-68 (2018) · Zbl 1405.65158
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