A time-splitting tau method for PDE’s: a contribution for the spectral Tau Toolbox library. (English) Zbl 07538974
Summary: This paper presents implementation details of an extension of the algebraic formulation for the spectral Tau method for the numerical solution of time-space partial differential problems, together with illustrative numerical examples. This extension implementation highlights (i) the orthogonal basis choice, (ii) the construction of the problem’s algebraic representation and (iii) the mechanisms to tackle certain partial differential problems with ease. This effort will be delivered to the scientific community as a crucial building block of the Tau Toolbox (a numerical library for the solution of integro-differential problems).
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
Software:
Tau ToolboxReferences:
| [1] | El-Daou, M.; Khajah, H., Iterated solutions of linear operator equations with the Tau method, Math. Comput., 66, 217, 207-213 (1997) · Zbl 0855.47006 · doi:10.1090/S0025-5718-97-00803-X |
| [2] | El Misiery, A.; Ortiz, E., Tau-lines: a new hybrid approach to the numerical treatment of crack problems based on the Tau method, Comput. Methods Appl. Mech. Eng., 56, 3, 265-282 (1986) · Zbl 0576.73095 · doi:10.1016/0045-7825(86)90042-3 |
| [3] | Hosseini, M.; Abadi, A.; Ortiz, E., A Tau method based on non-uniform space-time elements for the numerical simulation of solitons, Comput. Math. Appl., 22, 9, 7-19 (1991) · Zbl 0755.65124 · doi:10.1016/0898-1221(91)90202-F |
| [4] | Lanczos, C., Trigonometric interpolation of empirical and analytical functions, J. Math. Phys., 17, 1-4, 123-199 (1938) · Zbl 0020.01301 · doi:10.1002/sapm1938171123 |
| [5] | Matos, J.; Rodrigues, MJ; Vasconcelos, PB, New implementation of the Tau method for PDEs, J. Comput. Appl. Math., 164, 555-567 (2004) · Zbl 1038.65123 · doi:10.1016/j.cam.2003.09.054 |
| [6] | Namasivayam, S.; Ortiz, EL, Best approximation and the numerical solution of partial differential equations with the Tau method, Portugaliae mathematica, 40, 1, 97-119 (1981) · Zbl 0563.65061 |
| [7] | Onumanyi, P.; Ortiz, EL, Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the Tau method, Math. Comput., 43, 167, 189-203 (1984) · Zbl 0574.65091 · doi:10.1090/S0025-5718-1984-0744930-9 |
| [8] | Ortiz, E., Step by step Tau method—Part I. Piecewise polynomial approximations, Comput. Math. Appl., 1, 3, 381-392 (1975) · Zbl 0356.65006 · doi:10.1016/0898-1221(75)90040-1 |
| [9] | Ortiz, E.; Pun, K-S, Numerical solution of nonlinear partial differential equations with the Tau method, J. Comput. Appl. Math., 12-13, 511-516 (1985) · Zbl 0579.65124 · doi:10.1016/0377-0427(85)90044-5 |
| [10] | Ortiz, E.; Samara, H., Numerical solution of partial differential equations with variable coefficients with an operational approach to the Tau method, Comput. Math. Appl., 10, 1, 5-13 (1984) · Zbl 0575.65118 · doi:10.1016/0898-1221(84)90081-6 |
| [11] | Sarmin, E.; Chudov, L., On the stability of the numerical integration of systems of ordinary differential equations arising in the use of the straight line method, USSR Comput. Math. Math. Phys., 3, 6, 1537-1543 (1963) · Zbl 0189.49201 · doi:10.1016/0041-5553(63)90256-8 |
| [12] | Scheffel, J., A spectral method in time for initial-value problems, Am. J. Comput. Math., 2, 3, 173-193 (2012) · doi:10.4236/ajcm.2012.23023 |
| [13] | Verwer, J.; Sanz-Serna, J., Convergence of method of lines approximations to partial differential equations, Computing, 33, 3-4, 297-313 (1984) · Zbl 0546.65064 · doi:10.1007/BF02242274 |
| [14] | Wang, H., A time-splitting spectral method for coupled Gross-Pitaevskii equations with applications to rotating Bose-Einstein condensates, J. Comput. Appl. Math., 205, 1, 88-104 (2007) · Zbl 1118.65112 · doi:10.1016/j.cam.2006.04.042 |
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