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A fast ADI orthogonal spline collocation method with graded meshes for the two-dimensional fractional integro-differential equation. (English) Zbl 07400546

Summary: We propose and analyze a time-stepping Crank-Nicolson(CN) alternating direction implicit(ADI) scheme combined with an arbitrary-order orthogonal spline collocation (OSC) methods in space for the numerical solution of the fractional integro-differential equation with a weakly singular kernel. We prove the stability of the numerical scheme and derive error estimates. The analysis presented allows variable time steps which, as will be shown, can efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term. Finally, some numerical tests are given.

MSC:

65-XX Numerical analysis
35R11 Fractional partial differential equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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[1] Chen, H.; Xu, D.; Zhou, J., A second-order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel, J. Comput. Appl. Math., 356, 152-163 (2019) · Zbl 07069130
[2] Xu, D., On the discretization in time for a parabolic integro-differential equation with a weakly singular kernel, I: Smooth initial data, Appl. Math. Comput., 58, 1-27 (1993) · Zbl 0782.65160
[3] Chen, C.; Thomée, V.; Wahlbin, B., Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comput., 58, 587-602 (1992) · Zbl 0766.65120
[4] McLean, W.; Mustapha, K., A second-order accurate numerical method for a fractional wave equation, Numer. Math., 105, 481-510 (2007) · Zbl 1111.65113
[5] Larsson, S.; Thomée, V.; Wahlbin, B., Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comput. Amer Math. Soci., 67, 45-71 (1998) · Zbl 0896.65090
[6] Garrappa, R., Trapezoidal methods for fractional differential equations: theoretical and computational aspects, Math. Comput. Simul., 110, 96-112 (2015) · Zbl 07313349
[7] Qiao, L.; Xu, D., Compact alternating direction implicit scheme for integro-differential equations of parabolic type, J. Sci. Comput., 76, 565-582 (2018) · Zbl 1445.65050
[8] Qiao, L.; Xu, D., Orthogonal spline collocation scheme for the multi-term time-fractional diffusion equation, Int. J. Comput. Math., 95, 1478-1493 (2017)
[9] Qiao, L.; Xu, D.; Wang, Z.; An, ADI, Difference scheme based on fractional trapezoidal rule for fractional integro-differential equation with a weakly singular kernel, Appl. Math. Comput., 354, 103-114 (2019) · Zbl 1417.49030
[10] Qiao, L.; Xu, D., BDF ADI Orthogonal spline collocation scheme for the fractional integro-differential equation with two weakly singular kernels, Comput. Math. Appl., 78, 3807-3820 (2019) · Zbl 1443.65250
[11] Zhang, Y.; Sun, Z., Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230, 8713-8728 (2011) · Zbl 1242.65174
[12] Bialecki, B.; Fernandes, R., Alternating direction implicit orthogonal spline collocation on some non-rectangular regions with inconsistent partitions, Numer. Algor., 74, 1083-1100 (2017) · Zbl 1362.65111
[13] Yang, X.; Zhang, H.; Xu, D., Alternatting direction implicit OSC scheme for the two-dimensional fractional evolution equation with a weakly singular kernel, Acta Math. Sci., 38, 1689-1711 (2018) · Zbl 1399.35242
[14] Brunner, H., The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comp., 45, 417-437 (1985) · Zbl 0584.65093
[15] Chen, H.; Xu, D.; Peng, Y., A second order BDF alternating direction implicit difference scheme for the two dimensional fractional evolution equation, Appl. Math. Model., 41, 54-67 (2017) · Zbl 1443.65439
[16] Pani, A.; Fairweather, G.; Fernandes, R., Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term, SIAM J. Numer. Anal., 46, 344-364 (2008) · Zbl 1160.65068
[17] Pani, A.; Fairweather, G.; Fernandes, R., Orthogonal spline collocation methods for partial integro-differential equations, SIAM J. Numer. Anal., 30, 248-276 (2010) · Zbl 1191.65186
[18] Bialecki, B.; Fernandes, R., An orthogonal spline collocation alternating direction implicit Crank-Nicolson method for linear parabolic problems on rectangles, SIAM J. Numer. Anal., 36, 1414-1434 (1999) · Zbl 0955.65073
[19] Fairweather, G.; Yang, X.; Xu, D.; Zhang, H., An ADI Crank-Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion wave equation, J. Sci. Comput., 65, 1217-1239 (2015) · Zbl 1328.65216
[20] Gao, G.; Sun, Z., Two alternating direction implicit difference schemes for solving the two-dimensional time distributed-order wave equations, J. Sci. Comput., 69, 1-26 (2016) · Zbl 1372.65230
[21] Chen, S.; Liu, F., ADI-Euler and extrapolation methods for the two-dimensional advection-dispersion equation, J. Appl. Math. Comp., 26, 295-311 (2008) · Zbl 1146.76037
[22] Fairweather, G., Spline collocation methods for a class of hyperbolic partial integro-differential equations, SIAM J. Numer. Anal., 31, 444-460 (1994) · Zbl 0814.65137
[23] López-Marcos, J., A difference scheme for a nonlinear partial integro-differential equation, SIAM J. Numer. Anal., 27, 20-31 (1990) · Zbl 0693.65097
[24] Douglas, J., Jr.: Dupont, Collocation methods for parabolic equations in a single space variable, Lect. Notes Math., vol. 385. New York Springer (1974)
[25] Fernandes, R.; Fairweather, G., Analysis of alternating direction collocation methods for parabolic and hyperbolic problems in two space variables, Numer. Methods Partial Differ. Equ., 9, 191-211 (1993) · Zbl 0768.65067
[26] Yang, X.; Zhang, H.; Xu, D., WSGD-OSC Scheme for two-dimensional distributed order fractional reaction-diffusion equation, J. Sci. Comput., 76, 1502-1520 (2018) · Zbl 1397.65210
[27] Qiu, W., Xu, D., Guo, J.: A formally second-order BDF Sinc-collocation method for the Volterra integro-differential equation with a weakly singular kernel based on the double exponential transformation. Meth. Part Differ Equ. doi:10.1002/num.22703 (2020)
[28] Qiu, W.; Xu, D.; Guo, J., The Crank-Nicolson-type Sinc-Galerkin method for the fourth-order partial integro-differential equation with a weakly singular kernel, Appl. Numer. Math., 159, 239-258 (2021) · Zbl 1459.65189
[29] Yi, L.; Guo, B., An h-p version of the continuous Petrov-Galerkin finite element method for Volterra integro-differential equations with smooth and non-smooth kernels, SIAM J. Numer. Anal., 53, 2677-2704 (2015) · Zbl 1330.65206
[30] Wang, Z.; Guo, Y.; Yi, L., An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels, Math. Comp., 86, 2285-2324 (2017) · Zbl 1364.65299
[31] Fairweather, G.; Gladwell, I., Algorithms for almost block diagonal linear systems, SIAM Rev., 46, 49-58 (2004) · Zbl 1062.65031
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