A method of numerical conformal mapping of bounded regions with a rectilinear slit

Dongyi Li; Yibin Lu; Dongyi Li; Yibin Lu

doi:10.3934/math.2025388

AIMS Mathematics

2025, Volume 10, Issue 4: 8422-8445. doi: 10.3934/math.2025388

Previous Article Next Article

Research article

A method of numerical conformal mapping of bounded regions with a rectilinear slit

Dongyi Li ,
Yibin Lu ^,

Department of Mathematics, Kunming University of Science and Technology, Kunming 650500, Yunnan, China

Received: 06 January 2025 Revised: 24 March 2025 Accepted: 02 April 2025 Published: 14 April 2025
MSC : 65E10, 30-08, 76-10

This paper presented a numerical method based on the charge simulation method for calculating conformal mappings from the bounded multiply connected regions with a rectilinear slit onto the first category canonical slit domains. First, we proposed utilizing a pre-map function to expand a rectilinear slit in the bounded multiply connected regions. Second, to address pathologically constrained equation systems, we proposed a conjugate gradient squared (CGS) method combined with LU decomposition. Third, the conformal mappings were applied to simulate spiral point vortex bypass flow in regions with a rectilinear slit. Finally, numerical experiments validated the correctness of the mapping, the method's superiority, and its effectiveness in spiral point vortex bypass flow simulations.
- conformal mapping,
- charge simulation method,
- pre-map function,
- LU-CGS method,
- spiral point vortex
Citation: Dongyi Li, Yibin Lu. A method of numerical conformal mapping of bounded regions with a rectilinear slit[J]. AIMS Mathematics, 2025, 10(4): 8422-8445. doi: 10.3934/math.2025388

Related Papers:

Abstract

This paper presented a numerical method based on the charge simulation method for calculating conformal mappings from the bounded multiply connected regions with a rectilinear slit onto the first category canonical slit domains. First, we proposed utilizing a pre-map function to expand a rectilinear slit in the bounded multiply connected regions. Second, to address pathologically constrained equation systems, we proposed a conjugate gradient squared (CGS) method combined with LU decomposition. Third, the conformal mappings were applied to simulate spiral point vortex bypass flow in regions with a rectilinear slit. Finally, numerical experiments validated the correctness of the mapping, the method's superiority, and its effectiveness in spiral point vortex bypass flow simulations.

References

[1]	C. Shen, S. Mao, B. Xu, Z. Wang, X. Zhang, S. Yan, et al., Spiral complete coverage path planning based on conformal slit mapping in multi-connected domains, Int. J. Robot. Res., 43 (2024), 2183–2203. http://doi.org/10.1177/02783649241251385 doi: 10.1177/02783649241251385
[2]	X. D. Gu, W. Zeng, F. Luo, S.-T. Yau, Numerical computation of surface conformal mappings, Comput. Meth. Funct. Th., 11 (2012), 747–787. http://doi.org/10.1007/bf03321885 doi: 10.1007/bf03321885
[3]	P. Koebe, Abhandlungen zur theorie der konformen abbildung: Ⅳ. Abbildung mehrfach zusammenhängender schlichter bereiche auf schlitzbereiche, Acta Math., 41 (1916), 305–344. http://doi.org/10.1007/bf02422949 doi: 10.1007/bf02422949
[4]	G. T. Symm, An integral equation method in conformal mapping, Numer. Math., 9 (1966), 250–258. http://doi.org/10.1007/bf02162088 doi: 10.1007/bf02162088
[5]	D. Crowdy, J. Marshall, Conformal mappings between canonical multiply connected domains, Comput. Meth. Funct. Th., 6 (2006), 59–76. http://doi.org/10.1007/bf03321118 doi: 10.1007/bf03321118
[6]	K. Amano, A charge simulation method for numerical conformal mapping onto circular and radial slit domains, SIAM J. Sci. Comput., 19 (1998), 1169–1187. http://doi.org/10.1137/s1064827595294307 doi: 10.1137/s1064827595294307
[7]	K. Amano, D. Okano, Numerical conformal mappings onto the canonical slit domains, Theoretical and Applied Mechanics Japan, 60 (2012), 317–332. https://doi.org/10.11345/nctam.60.317 doi: 10.11345/nctam.60.317
[8]	K. Amano, D. Okano, A circular and radial slit mapping of unbounded multiply connected domains, JSIAM Lett., 2 (2010), 53–56. http://doi.org/10.14495/jsiaml.2.53 doi: 10.14495/jsiaml.2.53
[9]	D. Okano, H. Ogata, K. Amano, M. Sugihara, Numerical conformal mappings of bounded multiply connected domains by the charge simulation method, J. Comput. Appl. Math., 159 (2003), 109–117. http://doi.org/10.1016/s0377-0427(03)00572-7 doi: 10.1016/s0377-0427(03)00572-7
[10]	R. Wegmann, A. H. M. Murid, M. M. S. Nasser, The Riemann–Hilbert problem and the generalized Neumann kernel, J. Comput. Appl. Math., 182 (2005), 388–415. http://doi.org/10.1016/j.cam.2004.12.019 doi: 10.1016/j.cam.2004.12.019
[11]	M. M. S. Nasser, A boundary integral equation for conformal mapping of bounded multiply connected regions, Comput. Meth. Funct. Th., 9 (2009), 127–143. http://doi.org/10.1007/bf03321718 doi: 10.1007/bf03321718
[12]	A. W. K. Sangawi, A. H. M. Murid, M. M. S. Nasser, Linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits, Appl. Math. Comput., 218 (2011), 2055–2068. http://doi.org/10.1016/j.amc.2011.07.018 doi: 10.1016/j.amc.2011.07.018
[13]	M. M. S. Nasser, Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. Sci. Comput., 31 (2009), 1695–1715. http://doi.org/10.1137/070711438 doi: 10.1137/070711438
[14]	M. M. S. Nasser, Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe's canonical slit domains, J. Math. Anal. Appl., 382 (2011), 47–56. http://doi.org/10.1016/j.jmaa.2011.04.030 doi: 10.1016/j.jmaa.2011.04.030
[15]	M. M. S. Nasser, Numerical conformal mapping of multiply connected regions onto the fifth category of Koebe's canonical slit regions, J. Math. Anal. Appl., 398 (2013), 729–743. http://doi.org/10.1016/j.jmaa.2012.09.020 doi: 10.1016/j.jmaa.2012.09.020
[16]	Y. Nakatsukasa, O. Sète, L. N. Trefethen, The AAA algorithm for rational approximation, SIAM J. Sci. Comput., 40 (2018), A1494–A1522. https://doi.org/10.1137/16m1106122 doi: 10.1137/16m1106122
[17]	Y. Nakatsukasa, L. N. Trefethen, An algorithm for real and complex rational minimax approximation, SIAM J. Sci. Comput., 42 (2020), A3157–A3179. https://doi.org/10.1137/19m1281897 doi: 10.1137/19m1281897
[18]	L. N. Trefethen, Numerical conformal mapping with rational functions, Comput. Meth. Funct. Th., 20 (2020), 369–387. http://doi.org/10.1007/s40315-020-00325-w doi: 10.1007/s40315-020-00325-w
[19]	D. Okano, H. Ogata, K. Amano, A method of numerical conformal mapping of curved slit domains by the charge simulation method, J. Comput. Appl. Math., 152 (2003), 441–450. http://doi.org/10.1016/s0377-0427(02)00722-7 doi: 10.1016/s0377-0427(02)00722-7
[20]	H. A. Van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 13 (1992), 631–644. http://doi.org/10.1137/0913035 doi: 10.1137/0913035
[21]	P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 10 (1989), 36–52. http://doi.org/10.1137/0910004 doi: 10.1137/0910004
[22]	Z. Yang, Adaptive stochastic conjugate gradient for machine learning, Expert Syst. Appl., 206 (2022), 117719. https://doi.org/10.1016/j.eswa.2022.117719 doi: 10.1016/j.eswa.2022.117719
[23]	Z. Chen, W. Wang, X. Kong, L. Deng, Moving force identification based on the nonnegative flexible conjugate gradient least square method and experimental verification, J. Sound Vib., 572 (2024), 118177. https://doi.org/10.1016/j.jsv.2023.118177 doi: 10.1016/j.jsv.2023.118177
[24]	K. Murota, Comparison of conventional and "invariant" schemes of fundamental solutions method for annular domains, Japan J. Indust. Appl. Math., 12 (1995), 61–85. https://doi.org/10.1007/bf03167382 doi: 10.1007/bf03167382
[25]	Z. Nehari, Conformal mapping, Dover Publications, 2012.
[26]	J. Lin, V. Cevher, Kernel conjugate gradient methods with random projections, Appl. Comput. Harmon. Anal., 55 (2021), 223–269. https://doi.org/10.1016/j.acha.2021.05.004 doi: 10.1016/j.acha.2021.05.004
[27]	G. K. Batchelor, An introduction to fluid dynamics, Cambridge university press, 2000. https://doi.org/10.1017/CBO9780511800955
[28]	L. M. Milne-Thomson, Theoretical hydrodynamics, Dover Publications, 2011.
[29]	K. Amano, A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains, J. Comput. Appl. Math., 53 (1994), 353–370. https://doi.org/10.1016/0377-0427(94)90063-9 doi: 10.1016/0377-0427(94)90063-9

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)