Convergence of a proposed adaptive WENO scheme for Hamilton-Jacobi equations

Wonho Han; Kwangil Kim; Unhyok Hong

Applications of Mathematics (2023)

  • Volume: 68, Issue: 5, page 661-684
  • ISSN: 0862-7940

Abstract

top
We study high-order numerical methods for solving Hamilton-Jacobi equations. Firstly, by introducing new clear concise nonlinear weights and improving their convex combination, we develop WENO schemes of Zhu and Qiu (2017). Secondly, we give an algorithm of constructing a convergent adaptive WENO scheme by applying the simple adaptive step on the proposed WENO scheme, which is based on the introduction of a new singularity indicator. Through detailed numerical experiments on extensive problems including nonconvex ones, the convergence and effectiveness of the adaptive WENO scheme are demonstrated.

How to cite

top

Han, Wonho, Kim, Kwangil, and Hong, Unhyok. "Convergence of a proposed adaptive WENO scheme for Hamilton-Jacobi equations." Applications of Mathematics 68.5 (2023): 661-684. <http://eudml.org/doc/299520>.

@article{Han2023,
abstract = {We study high-order numerical methods for solving Hamilton-Jacobi equations. Firstly, by introducing new clear concise nonlinear weights and improving their convex combination, we develop WENO schemes of Zhu and Qiu (2017). Secondly, we give an algorithm of constructing a convergent adaptive WENO scheme by applying the simple adaptive step on the proposed WENO scheme, which is based on the introduction of a new singularity indicator. Through detailed numerical experiments on extensive problems including nonconvex ones, the convergence and effectiveness of the adaptive WENO scheme are demonstrated.},
author = {Han, Wonho, Kim, Kwangil, Hong, Unhyok},
journal = {Applications of Mathematics},
keywords = {Hamilton-Jacobi equations; WENO scheme; adaptive WENO scheme; nonconvex Hamiltonian; convergence},
language = {eng},
number = {5},
pages = {661-684},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Convergence of a proposed adaptive WENO scheme for Hamilton-Jacobi equations},
url = {http://eudml.org/doc/299520},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Han, Wonho
AU - Kim, Kwangil
AU - Hong, Unhyok
TI - Convergence of a proposed adaptive WENO scheme for Hamilton-Jacobi equations
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 5
SP - 661
EP - 684
AB - We study high-order numerical methods for solving Hamilton-Jacobi equations. Firstly, by introducing new clear concise nonlinear weights and improving their convex combination, we develop WENO schemes of Zhu and Qiu (2017). Secondly, we give an algorithm of constructing a convergent adaptive WENO scheme by applying the simple adaptive step on the proposed WENO scheme, which is based on the introduction of a new singularity indicator. Through detailed numerical experiments on extensive problems including nonconvex ones, the convergence and effectiveness of the adaptive WENO scheme are demonstrated.
LA - eng
KW - Hamilton-Jacobi equations; WENO scheme; adaptive WENO scheme; nonconvex Hamiltonian; convergence
UR - http://eudml.org/doc/299520
ER -

References

top
  1. Abgrall, R., 10.1137/040615997, SIAM J. Sci. Comput. 31 (2009), 2419-2446. (2009) Zbl1197.65167MR2520283DOI10.1137/040615997
  2. Amat, S., Ruiz, J., Shu, C.-W., 10.1137/18M1214937, SIAM J. Numer. Anal. 57 (2019), 1205-1237. (2019) Zbl1436.65095MR3956155DOI10.1137/18M1214937
  3. Bokanowski, O., Falcone, M., Sahu, S., 10.1137/140998482, SIAM J. Sci. Comput. 38 (2016), A171--A195. (2016) Zbl1407.65093MR3449908DOI10.1137/140998482
  4. Bryson, S., Levy, D., 10.1137/S0036142902408404, SIAM J. Numer. Anal. 41 (2003), 1339-1369. (2003) Zbl1050.65076MR2034884DOI10.1137/S0036142902408404
  5. Carlini, E., Ferretti, R., Russo, G., 10.1137/040608787, SIAM J. Sci. Comput. 27 (2005), 1071-1091. (2005) Zbl1105.65090MR2199921DOI10.1137/040608787
  6. Crandall, M. G., Lions, P.-L., 10.2307/2007396, Math. Comput. 43 (1984), 1-19. (1984) Zbl0556.65076MR0744921DOI10.2307/2007396
  7. Gottlieb, S., Shu, C.-W., Tadmor, E., 10.1137/S003614450036757X, SIAM Rev. 43 (2001), 89-112. (2001) Zbl0967.65098MR1854647DOI10.1137/S003614450036757X
  8. Henrick, A. K., Aslam, T. D., Powers, J. M., 10.1016/j.jcp.2005.01.023, J. Comput. Phys. 207 (2005), 542-567. (2005) Zbl1072.65114DOI10.1016/j.jcp.2005.01.023
  9. Huang, C., 10.1016/j.amc.2016.05.022, Appl. Math. Comput. 290 (2016), 21-32. (2016) Zbl1410.65313MR3523409DOI10.1016/j.amc.2016.05.022
  10. Jiang, G.-S., Peng, D., 10.1137/S106482759732455X, SIAM J. Sci. Comput. 21 (2000), 2126-2143. (2000) Zbl0957.35014MR1762034DOI10.1137/S106482759732455X
  11. Jiang, G.-S., Shu, C.-W., 10.1006/jcph.1996.0130, J. Comput. Phys. 126 (1996), 202-228. (1996) Zbl0877.65065MR1391627DOI10.1006/jcph.1996.0130
  12. Kim, K., Hong, U., Ri, K., Yu, J., 10.21136/AM.2021.0368-19, Appl. Math., Praha 66 (2021), 599-617. (2021) Zbl07396169MR4283305DOI10.21136/AM.2021.0368-19
  13. Kim, K., Li, Y., 10.1007/s10915-014-9955-5, J. Sci. Comput. 65 (2015), 110-137. (2015) Zbl1408.65053MR3394440DOI10.1007/s10915-014-9955-5
  14. Kurganov, A., Petrova, G., 10.1007/s10915-005-9033-0, J. Sci. Comput. 27 (2006), 323-333. (2006) Zbl1115.65093MR2285784DOI10.1007/s10915-005-9033-0
  15. Levy, D., Nayak, S., Shu, C.-W., Zhang, Y.-T., 10.1137/040612002, SIAM J. Sci. Comput. 28 (2006), 2229-2247. (2006) Zbl1126.65075MR2272259DOI10.1137/040612002
  16. Liu, X.-D., Osher, S., Chan, T., 10.1006/jcph.1994.1187, J. Comput. Phys. 115 (1994), 200-212. (1994) Zbl0811.65076MR1300340DOI10.1006/jcph.1994.1187
  17. Oberman, A. M., Salvador, T., 10.1016/j.jcp.2014.12.039, J. Comput. Phys. 284 (2015), 367-388. (2015) Zbl1352.65422MR3303624DOI10.1016/j.jcp.2014.12.039
  18. Osher, S., Shu, C.-W., 10.1137/0728049, SIAM J. Numer. Anal. 28 (1991), 907-922. (1991) Zbl0736.65066MR1111446DOI10.1137/0728049
  19. Qiu, J.-M., Shu, C.-W., 10.1137/070687487, SIAM J. Sci. Comput. 31 (2008), 584-607. (2008) Zbl1186.65123MR2460790DOI10.1137/070687487
  20. Qiu, J.-X., Shu, C.-W., 10.1016/j.jcp.2004.10.003, J. Comput. Phys. 204 (2005), 82-99. (2005) Zbl1070.65078MR2121905DOI10.1016/j.jcp.2004.10.003
  21. Shu, C.-W., 10.1137/070679065, SIAM Rev. 51 (2009), 82-126. (2009) Zbl1160.65330MR2481112DOI10.1137/070679065
  22. Xu, Z., Shu, C.-W., 10.4310/MAA.2005.v12.n2.a6, Methods Appl. Anal. 12 (2005), 169-190. (2005) Zbl1119.65378MR2257526DOI10.4310/MAA.2005.v12.n2.a6
  23. Zhang, Y.-T., Shu, C.-W., 10.1137/S1064827501396798, SIAM J. Sci. Comput. 24 (2003), 1005-1030. (2003) Zbl1034.65051MR1950522DOI10.1137/S1064827501396798
  24. Zhu, J., Qiu, J., 10.1016/j.jcp.2013.07.030, J. Comput. Phys. 254 (2013), 76-92. (2013) Zbl1349.65364MR3143358DOI10.1016/j.jcp.2013.07.030
  25. Zhu, J., Qiu, J., 10.1016/j.jcp.2016.05.010, J. Comput. Phys. 318 (2016), 110-121. (2016) Zbl1349.65365MR3503990DOI10.1016/j.jcp.2016.05.010
  26. Zhu, J., Qiu, J., 10.1002/num.22133, Numer. Methods Partial Differ. Equations 33 (2017), 1095-1113. (2017) Zbl1371.65089MR3652179DOI10.1002/num.22133

NotesEmbed ?

top

You must be logged in to post comments.