A superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels. (English) Zbl 1273.65198
The author studies a superconvergent discontinuous Galerkin method for Volterra integro-differential equations with smooth and non-smooth kernels. He first introduces the discontinuous Galerkin (DG) time-stepping method with a fixed approximation degree \(p\) on non-uniformly refined time-steps with \(p\geq1\). Later, he provides a global formulation of the DG scheme, introduces some projection operators and provides some technical lemmas and error estimates when \(\alpha\in(0,1)\). Also, he presents numerical examples to validate and illustrate the theoretical results discussed.
Reviewer: Seenith Sivasundaram (Daytona Beach)
MSC:
| 65R20 | Numerical methods for integral equations |
| 45D05 | Volterra integral equations |
| 45J05 | Integro-ordinary differential equations |
Keywords:
weakly singular kernel; smooth kernel; variable time steps; discontinuous Galerkin method; Volterra integro-differential equations; time-stepping method; error estimates; numerical examplesReferences:
| [1] | Hermann Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge Monographs on Applied and Computational Mathematics, vol. 15, Cambridge University Press, Cambridge, 2004. · Zbl 1059.65122 |
| [2] | Hermann Brunner, Arvet Pedas, and Gennadi Vainikko, Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal. 39 (2001), no. 3, 957 – 982. · Zbl 0998.65134 · doi:10.1137/S0036142900376560 |
| [3] | Hermann Brunner and Dominik Schötzau, \?\?-discontinuous Galerkin time-stepping for Volterra integrodifferential equations, SIAM J. Numer. Anal. 44 (2006), no. 1, 224 – 245. · Zbl 1116.65127 · doi:10.1137/040619314 |
| [4] | M. Delfour, W. Hager, and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp. 36 (1981), no. 154, 455 – 473. · Zbl 0469.65053 |
| [5] | Kenneth Eriksson, Claes Johnson, and Vidar Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 4, 611 – 643 (English, with French summary). · Zbl 0589.65070 |
| [6] | Donald Estep, A posteriori error bounds and global error control for approximation of ordinary differential equations, SIAM J. Numer. Anal. 32 (1995), no. 1, 1 – 48. · Zbl 0820.65052 · doi:10.1137/0732001 |
| [7] | Ying-Jun jiang, On spectral methods for Volterra-type integro-differential equations, J. Comput. Appl. Math. 230 (2009), no. 2, 333 – 340. · Zbl 1202.65170 · doi:10.1016/j.cam.2008.12.001 |
| [8] | Claes Johnson, Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations, SIAM J. Numer. Anal. 25 (1988), no. 4, 908 – 926. · Zbl 0661.65076 · doi:10.1137/0725051 |
| [9] | Stig Larsson, Vidar Thomée, and Lars B. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comp. 67 (1998), no. 221, 45 – 71. · Zbl 0896.65090 |
| [10] | P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89 – 123. Publication No. 33. · Zbl 0341.65076 |
| [11] | William McLean and Kassem Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math. 105 (2007), no. 3, 481 – 510. · Zbl 1111.65113 · doi:10.1007/s00211-006-0045-y |
| [12] | K. Mustapha, A Petrov-Galerkin method for integro-differential equations with a memory term, ANZIAM J. 50 (2008), no. (C), C610 – C624. |
| [13] | Kassem Mustapha and William McLean, Discontinuous Galerkin method for an evolution equation with a memory term of positive type, Math. Comp. 78 (2009), no. 268, 1975 – 1995. · Zbl 1198.65195 |
| [14] | K. Mustapha, H. Brunner, H. Mustapha, and D. Schötzau, An \?\?-version discontinuous Galerkin method for integro-differential equations of parabolic type, SIAM J. Numer. Anal. 49 (2011), no. 4, 1369 – 1396. · Zbl 1230.65143 · doi:10.1137/100797114 |
| [15] | Kassem Mustapha and Hussein Mustapha, A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel, IMA J. Numer. Anal. 30 (2010), no. 2, 555 – 578. · Zbl 1193.65225 · doi:10.1093/imanum/drn075 |
| [16] | Amiya Kumar Pani, Graeme Fairweather, and Ryan I. Fernandes, ADI orthogonal spline collocation methods for parabolic partial integro-differential equations, IMA J. Numer. Anal. 30 (2010), no. 1, 248 – 276. · Zbl 1191.65186 · doi:10.1093/imanum/drp024 |
| [17] | Amiya K. Pani and Sangita Yadav, An \?\?-local discontinuous Galerkin method for parabolic integro-differential equations, J. Sci. Comput. 46 (2011), no. 1, 71 – 99. · Zbl 1227.65133 · doi:10.1007/s10915-010-9384-z |
| [18] | W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory, LA-UR-73-479, 1973. |
| [19] | Dominik Schötzau and Christoph Schwab, Time discretization of parabolic problems by the \?\?-version of the discontinuous Galerkin finite element method, SIAM J. Numer. Anal. 38 (2000), no. 3, 837 – 875. · Zbl 0978.65091 · doi:10.1137/S0036142999352394 |
| [20] | D. Schötzau and C. Schwab, An \?\? a priori error analysis of the DG time-stepping method for initial value problems, Calcolo 37 (2000), no. 4, 207 – 232. · Zbl 1012.65084 · doi:10.1007/s100920070002 |
| [21] | T. Tang, A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal. 13 (1993), no. 1, 93 – 99. · Zbl 0765.65126 · doi:10.1093/imanum/13.1.93 |
| [22] | Tao Tang, Xiang Xu, and Jin Cheng, On spectral methods for Volterra integral equations and the convergence analysis, J. Comput. Math. 26 (2008), no. 6, 825 – 837. · Zbl 1174.65058 |
| [23] | Y. Wei and Y. 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. · Zbl 1262.45005 |
| [24] | Vidar Thomée, Galerkin finite element methods for parabolic problems, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. · Zbl 1105.65102 |
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