A spatially sixth-order hybrid L 1 -CCD method for solving time fractional Schrödinger equations

Chun-Hua Zhang; Jun-Wei Jin; Hai-Wei Sun; Qin Sheng

Applications of Mathematics (2021)

  • Volume: 66, Issue: 2, page 213-232
  • ISSN: 0862-7940

Abstract

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We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an L 1 strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid L 1 -CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order 2 - γ in time, where 0 < γ < 1 is the order of the Caputo fractional derivative involved. It is proved rigorously that the hybrid numerical method accomplished is unconditionally stable in the Fourier sense. Numerical experiments are carried out with typical testing problems to validate the effectiveness of the new algorithms.

How to cite

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Zhang, Chun-Hua, et al. "A spatially sixth-order hybrid $L1$-CCD method for solving time fractional Schrödinger equations." Applications of Mathematics 66.2 (2021): 213-232. <http://eudml.org/doc/297086>.

@article{Zhang2021,
abstract = {We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an $L1$ strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid $L1$-CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order $2-\gamma $ in time, where $0<\gamma <1$ is the order of the Caputo fractional derivative involved. It is proved rigorously that the hybrid numerical method accomplished is unconditionally stable in the Fourier sense. Numerical experiments are carried out with typical testing problems to validate the effectiveness of the new algorithms.},
author = {Zhang, Chun-Hua, Jin, Jun-Wei, Sun, Hai-Wei, Sheng, Qin},
journal = {Applications of Mathematics},
keywords = {nonlinear time fractional Schrödinger equations; $L1$ formula; hybrid compact difference method; linearization; unconditional stability},
language = {eng},
number = {2},
pages = {213-232},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A spatially sixth-order hybrid $L1$-CCD method for solving time fractional Schrödinger equations},
url = {http://eudml.org/doc/297086},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Zhang, Chun-Hua
AU - Jin, Jun-Wei
AU - Sun, Hai-Wei
AU - Sheng, Qin
TI - A spatially sixth-order hybrid $L1$-CCD method for solving time fractional Schrödinger equations
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 213
EP - 232
AB - We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an $L1$ strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid $L1$-CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order $2-\gamma $ in time, where $0<\gamma <1$ is the order of the Caputo fractional derivative involved. It is proved rigorously that the hybrid numerical method accomplished is unconditionally stable in the Fourier sense. Numerical experiments are carried out with typical testing problems to validate the effectiveness of the new algorithms.
LA - eng
KW - nonlinear time fractional Schrödinger equations; $L1$ formula; hybrid compact difference method; linearization; unconditional stability
UR - http://eudml.org/doc/297086
ER -

References

top
  1. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J., 10.1142/8180, Series on Complexity, Nonlinearity and Chaos 5. World Scientific, Hackensack (2012). (2012) Zbl1248.26011MR2894576DOI10.1142/8180
  2. Benson, D. A., Schumer, R., Meerschaert, M. M., Wheatcraft, S. W., 10.1023/A:1006733002131, Transp. Porous Media 42 (2001), 211-240. (2001) MR1948593DOI10.1023/A:1006733002131
  3. Bhrawy, A. H., Doha, E. H., Ezz-Eldien, S. S., Gorder, R. A. Van, 10.1140/epjp/i2014-14260-6, Eur. Phys. J. Plus 129 (2014), Article ID 260, 21 pages. (2014) MR2930392DOI10.1140/epjp/i2014-14260-6
  4. Bratsos, A. G., 10.1007/BF02941979, Korean J. Comput. Appl. Math. 8 (2001), 459-467. (2001) Zbl1015.65042MR1848910DOI10.1007/BF02941979
  5. Chang, Q., Jia, E., Sun, W., 10.1006/jcph.1998.6120, J. Comput. Phys. 148 (1999), 397-415. (1999) Zbl0923.65059MR1669707DOI10.1006/jcph.1998.6120
  6. Chen, X., Di, Y., Duan, J., Li, D., 10.1016/j.aml.2018.05.007, Appl. Math. Lett. 84 (2018), 160-167. (2018) Zbl06892656MR3808513DOI10.1016/j.aml.2018.05.007
  7. Chen, B., He, D., Pan, K., 10.4208/nmtma.OA-2017-0090, Numer. Math., Theory Methods Appl. 11 (2018), 299-320. (2018) Zbl1424.65124MR3849456DOI10.4208/nmtma.OA-2017-0090
  8. Chen, B., He, D., Pan, K., 10.1080/00207160.2018.1478415, Int. J. Comput. Math. 96 (2019), 992-1004. (2019) MR3911287DOI10.1080/00207160.2018.1478415
  9. Chu, P. C., Fan, C., 10.1006/jcph.1998.5899, J. Comput. Phys. 140 (1998), 370-399. (1998) Zbl0923.65071MR1616146DOI10.1006/jcph.1998.5899
  10. Cui, M., 10.1016/j.jcp.2009.07.021, J. Comput. Phys. 228 (2009), 7792-7804. (2009) Zbl1179.65107MR2561843DOI10.1016/j.jcp.2009.07.021
  11. Dehghan, M., Taleei, A., 10.1016/j.cpc.2009.08.015, Comput. Phys. Commun. 181 (2010), 43-51. (2010) Zbl1206.65207MR2575184DOI10.1016/j.cpc.2009.08.015
  12. Feynman, R. P., Hibbs, A. R., Quantum Mechanics and Path Integrals, Dover Publications, New York (2010). (2010) Zbl1220.81156MR2797644
  13. Gao, G.-H., Sun, H.-W., 10.4208/cicp.180314.010914a, Commun. Comput. Phys. 17 (2015), 487-509. (2015) Zbl1388.65052MR3371511DOI10.4208/cicp.180314.010914a
  14. Gao, G.-H., Sun, H.-W., 10.1016/j.jcp.2015.05.052, J. Comput. Phys. 298 (2015), 520-538. (2015) Zbl1349.65294MR3374563DOI10.1016/j.jcp.2015.05.052
  15. Gao, G.-H., Sun, H.-W., Sun, Z.-Z., 10.1016/j.jcp.2014.09.033, J. Comput. Phys. 280 (2015), 510-528. (2015) Zbl1349.65295MR3273149DOI10.1016/j.jcp.2014.09.033
  16. Gao, G.-H., Sun, Z.-Z., 10.1016/j.jcp.2010.10.007, J. Comput. Phys. 230 (2011), 586-595. (2011) Zbl1211.65112MR2745445DOI10.1016/j.jcp.2010.10.007
  17. Gao, G.-H., Sun, Z.-Z., Zhang, H.-W., 10.1016/j.jcp.2013.11.017, J. Comput. Phys. 259 (2014), 33-50. (2014) Zbl1349.65088MR3148558DOI10.1016/j.jcp.2013.11.017
  18. Golmankhaneh, A. K., Baleanu, D., 10.1515/9783110472097-019, Fractional Dynamics De Gruyter, Berlin (2015), 307-332. (2015) Zbl1333.00024MR3468175DOI10.1515/9783110472097-019
  19. He, D., 10.1007/s11075-015-0082-7, Numer. Algorithms 72 (2016), 1103-1117. (2016) Zbl1350.65088MR3529835DOI10.1007/s11075-015-0082-7
  20. He, D., Pan, K., 10.4208/eajam.200816.300517a, East Asian J. Appl. Math. 7 (2017), 714-727. (2017) Zbl1426.76471MR3766403DOI10.4208/eajam.200816.300517a
  21. He, D., Pan, K., 10.1016/j.camwa.2017.04.009, Comput. Math. Appl. 73 (2017), 2360-2374. (2017) Zbl1373.65056MR3648018DOI10.1016/j.camwa.2017.04.009
  22. Jones, T. N., Sheng, Q., 10.1016/j.jcp.2019.04.046, J. Comput. Phys. 392 (2019), 403-418. (2019) MR3948727DOI10.1016/j.jcp.2019.04.046
  23. Khan, N. A., Jamil, M., Ara, A., 10.5402/2012/197068, Int. Sch. Res. Not. 2012 (2012), Article ID 197068, 11 pages. (2012) DOI10.5402/2012/197068
  24. Klages, R., Radons, G., (eds.), I. M. Sokolov, 10.1002/9783527622979, Wiley, Weinheim (2008). (2008) DOI10.1002/9783527622979
  25. Laskin, N., 10.1016/S0375-9601(00)00201-2, Phys. Lett., A 268 (2000), 298-305. (2000) Zbl0948.81595MR1755089DOI10.1016/S0375-9601(00)00201-2
  26. Lee, S. T., Liu, J., Sun, H.-W., 10.1016/j.cam.2014.01.004, J. Comput. Appl. Math. 264 (2014), 23-37. (2014) Zbl1294.65098MR3164100DOI10.1016/j.cam.2014.01.004
  27. Li, L. Z., Sun, H.-W., Tam, S.-C., 10.1016/j.cpc.2014.10.008, Comput. Phys. Commun. 187 (2015), 38-48. (2015) Zbl1348.35238MR3283134DOI10.1016/j.cpc.2014.10.008
  28. Li, D., Wang, J., Zhang, J., 10.1137/16M1105700, SIAM. J. Sci. Comput. 39 (2017), A3067--A3088. (2017) Zbl1379.65079MR3738320DOI10.1137/16M1105700
  29. Liao, H.-L., Shi, H.-S., Zhao, Y., 10.1016/j.cpc.2014.05.002, Comput. Phys. Commun. 185 (2014), 2240-2249. (2014) Zbl1344.35137MR3211950DOI10.1016/j.cpc.2014.05.002
  30. Mohebbi, A., Abbaszadeh, M., Dehghan, M., 10.1016/j.enganabound.2012.12.002, Eng. Anal. Bound. Elem. 37 (2013), 475-485. (2013) Zbl1352.65397MR3018826DOI10.1016/j.enganabound.2012.12.002
  31. Murio, D. A., 10.1016/j.camwa.2008.02.015, Comput. Math. Appl. 56 (2008), 1138-1145. (2008) Zbl1155.65372MR2437884DOI10.1016/j.camwa.2008.02.015
  32. Naber, M., 10.1063/1.1769611, J. Math. Phys. 45 (2004), 3339-3352. (2004) Zbl1071.81035MR2077514DOI10.1063/1.1769611
  33. Rida, S. Z., El-Sherbiny, H. M., Arafa, A. A. M., 10.1016/j.physleta.2007.06.071, Phys. Lett., A 372 (2008), 553-558. (2008) Zbl1217.81068MR2378723DOI10.1016/j.physleta.2007.06.071
  34. Scalas, E., Gorenflo, R., Mainardi, F., 10.1016/S0378-4371(00)00255-7, Phys. A 284 (2000), 376-384. (2000) MR1773804DOI10.1016/S0378-4371(00)00255-7
  35. Shivanian, E., Jafarabadi, A., 10.1002/num.22126, Numer. Methods Partial Differ. Equations 33 (2017), 1043-1069. (2017) Zbl1379.65082MR3652177DOI10.1002/num.22126
  36. Sun, H.-W., Li, L. Z., 10.1016/j.cpc.2013.11.009, Comput. Phys. Commun. 185 (2014), 790-797. (2014) Zbl1360.35193MR3161142DOI10.1016/j.cpc.2013.11.009
  37. Sun, Z.-Z., Wu, X., 10.1016/j.apnum.2005.03.003, Appl. Numer. Math. 56 (2006), 193-209. (2006) Zbl1094.65083MR2200938DOI10.1016/j.apnum.2005.03.003
  38. Wang, Y.-M., Ren, L., 10.1080/00207160.2018.1437262, Int. J. Comput. Math. 96 (2019), 264-297. (2019) MR3893479DOI10.1080/00207160.2018.1437262
  39. Wang, Z., Vong, S., 10.1016/j.jcp.2014.08.012, J. Comput. Phys. 277 (2014), 1-15. (2014) Zbl1349.65348MR3254221DOI10.1016/j.jcp.2014.08.012
  40. Wei, L., He, Y., Zhang, X., Wang, S., 10.1016/j.finel.2012.03.008, Finite Elem. Anal. Des. 59 (2012), 28-34. (2012) MR2949417DOI10.1016/j.finel.2012.03.008
  41. Xu, Y., Zhang, L., 10.1016/j.cpc.2012.01.006, Comput. Phys. Commun. 183 (2012), 1082-1093. (2012) Zbl1277.65073MR2888993DOI10.1016/j.cpc.2012.01.006
  42. Zhu, L., Sheng, Q., 10.1016/j.cam.2020.112714, J. Comput. Appl. Math. 374 (2020), Article ID 112714, 14 pages. (2020) Zbl1435.65136MR4066721DOI10.1016/j.cam.2020.112714

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