A new energy conservative scheme for regularized long wave equation

Yuesheng Luo; Ruixue Xing; Xiaole Li

Applications of Mathematics (2021)

  • Volume: 66, Issue: 5, page 745-765
  • ISSN: 0862-7940

Abstract

top
An energy conservative scheme is proposed for the regularized long wave (RLW) equation. The integral method with variational limit is used to discretize the spatial derivative and the finite difference method is used to discretize the time derivative. The energy conservation of the scheme and existence of the numerical solution are proved. The convergence of the order O ( h 2 + τ 2 ) and unconditional stability are also derived. Numerical examples are carried out to verify the correctness of the theoretical analysis.

How to cite

top

Luo, Yuesheng, Xing, Ruixue, and Li, Xiaole. "A new energy conservative scheme for regularized long wave equation." Applications of Mathematics 66.5 (2021): 745-765. <http://eudml.org/doc/297618>.

@article{Luo2021,
abstract = {An energy conservative scheme is proposed for the regularized long wave (RLW) equation. The integral method with variational limit is used to discretize the spatial derivative and the finite difference method is used to discretize the time derivative. The energy conservation of the scheme and existence of the numerical solution are proved. The convergence of the order $O(h^2 + \tau ^2)$ and unconditional stability are also derived. Numerical examples are carried out to verify the correctness of the theoretical analysis.},
author = {Luo, Yuesheng, Xing, Ruixue, Li, Xiaole},
journal = {Applications of Mathematics},
keywords = {regularized long wave equation; integral method with variational limit; finite difference method; Lagrange interpolation; energy conservation scheme},
language = {eng},
number = {5},
pages = {745-765},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A new energy conservative scheme for regularized long wave equation},
url = {http://eudml.org/doc/297618},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Luo, Yuesheng
AU - Xing, Ruixue
AU - Li, Xiaole
TI - A new energy conservative scheme for regularized long wave equation
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 5
SP - 745
EP - 765
AB - An energy conservative scheme is proposed for the regularized long wave (RLW) equation. The integral method with variational limit is used to discretize the spatial derivative and the finite difference method is used to discretize the time derivative. The energy conservation of the scheme and existence of the numerical solution are proved. The convergence of the order $O(h^2 + \tau ^2)$ and unconditional stability are also derived. Numerical examples are carried out to verify the correctness of the theoretical analysis.
LA - eng
KW - regularized long wave equation; integral method with variational limit; finite difference method; Lagrange interpolation; energy conservation scheme
UR - http://eudml.org/doc/297618
ER -

References

top
  1. Brango, C. Banquet, 10.1016/j.physd.2011.10.007, Physica D 241 (2012), 125-133. (2012) Zbl1252.35130DOI10.1016/j.physd.2011.10.007
  2. Bhardwaj, D., Shankar, R., 10.1016/S0898-1221(00)00248-0, Comput. Math. Appl. 40 (2000), 1397-1404. (2000) Zbl0965.65108MR1803919DOI10.1016/S0898-1221(00)00248-0
  3. Bhowmik, S. K., Karakoc, S. B. G., 10.1002/num.22410, Numer. Methods Partial Differ. Equations 35 (2019), 2236-2257. (2019) Zbl1431.65169MR4022940DOI10.1002/num.22410
  4. Cai, J., 10.1016/j.cpc.2009.05.009, Comput. Phys. Commun. 180 (2009), 1821-1831. (2009) Zbl1197.65144MR2678455DOI10.1016/j.cpc.2009.05.009
  5. Chegini, N. G., Salaripanah, A., Mokhtari, R., Isvand, D., 10.1007/s11071-011-0277-y, Nonlinear Dyn. 69 (2012), 459-471. (2012) Zbl1258.65076MR2929885DOI10.1007/s11071-011-0277-y
  6. Chertovskih, R., Chian, A. C.-L., Podvigina, O., Rempel, E. L., Zheligovsky, V., Existence, uniqueness, and analyticity of space-periodic solutions to the regularized long-wave equation, Adv. Differ. Equ. 19 (2014), 725-754. (2014) Zbl1292.35227MR3252900
  7. Dogan, A., 10.1016/S0307-904X(01)00084-1, Appl. Math. Modelling 26 (2002), 771-783. (2002) Zbl1016.76046DOI10.1016/S0307-904X(01)00084-1
  8. Eilbeck, J. C., McGuire, G. R., 10.1016/0021-9991(75)90115-1, J. Comput. Phys. 19 (1975), 43-57. (1975) Zbl0325.65054MR0400907DOI10.1016/0021-9991(75)90115-1
  9. Fang, S., Guo, B., Qiu, H., 10.1016/j.cnsns.2007.07.001, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 61-68. (2009) Zbl1221.35362MR2458711DOI10.1016/j.cnsns.2007.07.001
  10. Gardner, L. R. T., Gardner, G. A., Dag, I., 10.1002/cnm.1640110109, Commun. Numer. Methods Eng. 11 (1995), 59-68. (1995) Zbl0819.65125MR1312879DOI10.1002/cnm.1640110109
  11. Guo, L., Chen, H., 10.1007/s00607-005-0158-7, Computing 77 (2006), 205-221. (2006) Zbl1098.65096MR2214448DOI10.1007/s00607-005-0158-7
  12. Guo, B., Shang, Y., 10.1007/s10255-003-0095-1, Acta Math. Appl. Sin., Engl. Ser. 19 (2003), 191-204. (2003) Zbl1059.35105MR2011482DOI10.1007/s10255-003-0095-1
  13. Hammad, D. A., El-Azab, M. S., 10.1016/j.amc.2016.03.033, Appl. Math. Comput. 285 (2016), 228-240. (2016) Zbl1410.65395MR3494425DOI10.1016/j.amc.2016.03.033
  14. Hu, J., Zheng, K., 10.1155/2010/543503, Bound. Value Probl. 2010 (2010), Article ID 543503, 18 pages. (2010) Zbl1187.65090MR2600713DOI10.1155/2010/543503
  15. Irk, D., Keskin, P., 10.11948/2017038, J. Appl. Anal. Comput. 7 (2017), 617-631. (2017) MR3602441DOI10.11948/2017038
  16. Irk, D., Yildiz, P. Keskin, Görgülü, M. Zorşahin, 10.3906/mat-1804-55, Turk. J. Math. 43 (2019), 112-125. (2019) Zbl1417.65172MR3909279DOI10.3906/mat-1804-55
  17. Karakoc, S. B. G., Yagmurlu, N. M., Ucar, Y., 10.1186/1687-2770-2013-27, Bound. Value Probl. 2013 (2013), Article ID 27, 17 pages. (2013) Zbl1284.65142MR3110753DOI10.1186/1687-2770-2013-27
  18. Kumar, R., Baskar, S., 10.1016/j.cam.2015.06.015, J. Comput. Appl. Math. 292 (2016), 41-66. (2016) Zbl1329.65236MR3392380DOI10.1016/j.cam.2015.06.015
  19. Lin, B., 10.1111/sapm.12022, Stud. Appl. Math. 132 (2014), 160-182. (2014) Zbl1291.65302MR3167092DOI10.1111/sapm.12022
  20. Lin, B., 10.1016/j.amc.2014.05.133, Appl. Math. Comput. 243 (2014), 358-367. (2014) Zbl1336.65176MR3244483DOI10.1016/j.amc.2014.05.133
  21. Lin, B., 10.1080/00207160.2014.950254, Int. J. Comput. Math. 92 (2015), 1591-1607. (2015) Zbl1317.65054MR3340634DOI10.1080/00207160.2014.950254
  22. Luo, Y., Li, X., Guo, C., 10.1002/num.22143, Numer. Methods Partial Differ. Equations 33 (2017), 1283-1304. (2017) Zbl1377.65119MR3652187DOI10.1002/num.22143
  23. Oruç, Ö., Bulut, F., Esen, A., 10.1007/s00009-016-0682-z, Mediterr. J. Math. 13 (2016), 3235-3253. (2016) Zbl1354.65194MR3554305DOI10.1007/s00009-016-0682-z
  24. Peregrine, D. H., 10.1017/S0022112066001678, J. Fluid Mech. 25 (1966), 321-330. (1966) DOI10.1017/S0022112066001678
  25. Peregrine, D. H., 10.1017/S0022112067002605, J. Fluid Mech. 27 (1967), 815-827. (1967) Zbl0163.21105DOI10.1017/S0022112067002605
  26. Pindza, E., Maré, E., 10.1155/2014/178024, Int. J. Comput. Math. 2014 (2014), Article ID 178024, 12 pages. (2014) DOI10.1155/2014/178024
  27. Raslan, K. R., 10.1016/j.amc.2004.06.130, Appl. Math. Comput. 167 (2005), 1101-1118. (2005) Zbl1082.65582MR2169754DOI10.1016/j.amc.2004.06.130
  28. Rouatbi, A., Achouri, T., Omrani, K., 10.1007/s40314-017-0567-1, Comput. Appl. Math. 37 (2018), 4169-4195. (2018) Zbl1402.65090MR3848530DOI10.1007/s40314-017-0567-1
  29. Salih, H., Tawfiq, L. N. M., Yahya, Z. R., Zin, S. Mat, 10.1088/1742-6596/1003/1/012062, J. Phys., Conf. Ser. 1003 (2018), Article ID 012062, 9 pages. (2018) DOI10.1088/1742-6596/1003/1/012062
  30. Shang, Y., Guo, B., 10.1007/BF02440077, Appl. Math. Mech., Engl. Ed. 26 (2005), 283-291. (2005) Zbl1144.76304MR2132120DOI10.1007/BF02440077
  31. Shao, X., Xue, G., Li, C., 10.1016/j.amc.2013.03.068, Appl. Math. Comput. 219 (2013), 9202-9209. (2013) Zbl1288.65125MR3047814DOI10.1016/j.amc.2013.03.068
  32. Soliman, A. A., 10.1080/00207160412331272135, Int. J. Comput. Math. 81 (2004), 1281-1288. (2004) Zbl1063.65086MR2173459DOI10.1080/00207160412331272135
  33. Wang, B., Sun, T., Liang, D., 10.1016/j.cam.2019.01.036, J. Comput. Appl. Math. 356 (2019), 98-117. (2019) Zbl1419.65033MR3915392DOI10.1016/j.cam.2019.01.036
  34. Xie, S., Kim, S., Woo, G., Yi, S., 10.1137/070683623, SIAM J. Sci. Comput. 30 (2008), 2263-2285. (2008) Zbl1181.65125MR2429465DOI10.1137/070683623
  35. Yan, J., Lai, M.-C., Li, Z., Zhang, Z., 10.4208/aamm.2014.m888, Adv. Appl. Math. Mech. 9 (2017), 250-271. (2017) MR3598526DOI10.4208/aamm.2014.m888
  36. Zhang, L., 10.1016/j.amc.2004.09.027, Appl. Math. Comput. 168 (2005), 962-972. (2005) Zbl1080.65079MR2171754DOI10.1016/j.amc.2004.09.027
  37. Zheng, K., Hu, J., 10.1186/1687-1847-2013-287, Adv. Difference Equ. 2013 (2013), Article ID 287, 12 pages. (2013) Zbl1444.65051MR3337283DOI10.1186/1687-1847-2013-287
  38. Zhou, Y., Applications of Discrete Functional Analysis to the Finite Difference Method, International Academic Publishers, Beijing (1991). (1991) Zbl0732.65080MR1133399

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.