Abstract
We propose a numerical method for the conformal mapping of unbounded multiply connected domains exterior to closed Jordan curves C 1, . . . ,C n onto a canonical linear slit domain, which is the entire plane with linear slits S 1, . . . , S n of angles θ 1, . . . , θ n arbitrarily assigned to the real axis, respectively. If θ 1 = · · · = θ n = θ then it is the well-known parallel slit domain, which is important in the problem of potential flows past obstacles. In the method, we reduce the mapping problem to a boundary value problem for an analytic function, and approximate it by a linear combination of complex logarithmic functions based on the charge simulation method. Numerical examples show the effectiveness of our method.
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Amano K.: Numerical conformal mapping based on the charge simulation method (in Japanese). Trans. Inform. Process. Soc. Japan 28, 697–704 (1987)
Amano K.: A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. J. Comput. Appl. Math. 53, 353–370 (1994)
Amano K.: A charge simulation method for numerical conformal mapping onto circular and radial slit domains. SIAM J. Sci. Comput. 19, 1169–1187 (1998)
Amano K., Okano D., Ogata H., Shimohira H., Sugihara M.: A systematic scheme of numerical conformal mappings of unbounded multiply-connected domains by the charge simulation method (in Japanese). IPSJ J. 42, 385–395 (2001)
Amano K., Ootori H., Li T., Endo K., Okano D.: Numerical conformal mappings onto a rectilinear slit domain by the charge simulation method (in Japanese). IPSJ J. 50, 1775–1779 (2009)
Amano K., Okano D.: A circular and radial slit mapping of unbounded multiply connected domains. JSIAM Lett. 2, 53–56 (2010)
Benchama N., DeLillo T.K., Hrycak T., Wang L.: A simplified Fornberg-like method for the conformal mapping of multiply connected regions—comparisons and crowding. J. Comput. Appl. Math. 209, 1–21 (2007)
Crowdy D.G.: Analytical solutions for uniform potential flow past multiple cylinders. Eur. J. Mech. B Fluids 25, 459–470 (2006)
Crowdy D.G., Marshall J.: Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6, 59–76 (2006)
DeLillo T.K., Horn M.A., Pfaltzgraff J.A.: Numerical conformal mapping of multiply connected regions by Fornberg-like methods. Numer. Math. 83, 205–230 (1999)
DeLillo T.K., Elcrat A.R., Pfaltzgraff J.A.: Schwarz–Christoffel mapping of multiply connected domains. J. Anal. Math. 94, 17–47 (2004)
DeLillo T.K., Driscoll T.A., Elcrat A.R., Pfaltzgraff J.A.: Radial and circular slit maps of unbounded multiply connected circle domains. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464, 1719–1737 (2008)
Driscoll T.A., Trefethen L.N.: Schwarz–Christoffel Mapping. Cambridge University Press, Cambridge (2002)
Fornberg B.: A numerical method for conformal mappings. SIAM J. Sci. Stat. Comput. 1, 386–400 (1980)
Fornberg B.: A numerical method for conformal mapping of doubly connected regions. SIAM J. Sci. Stat. Comput. 5, 771–783 (1984)
Gaier D.: Konstruktive Methoden der konformen Abbildung. Springer, Berlin (1964)
Gaier D.: Integralgleichungen erster Art und konforme Abbildung. Math. Z. 147, 113–129 (1976)
Gaier, D.: Das logarithmische Potential und die konforme Abbildung mehrfach zusammenhängender Gebiete. In: Butzer, P.L., Fehér, F. (eds.) E.B. Christoffel, The Influence of his Work on Mathematics and the Physical Scienecs, pp. 290–303. Birkhäuser, Basel (1981)
Gutknecht M.H.: Numerical conformal mapping methods based on function conjugation. J. Comput. Appl. Math. 14, 31–77 (1986)
Hayes J.K., Kahaner D.K., Kellner R.G.: An improved method for numerical conformal mapping. Math. Comput. 26, 327–334 (1972)
Henrici P.: Applied and Computational Complex Analysis, vol. 3. Wiley, New York (1986)
Hough D.M., Papamichael N.: The use of splines and singular functions in an integral equation method for conformal mapping. Numer. Math. 37, 133–147 (1981)
Hough D.M., Papamichael N.: An integral equation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. Numer. Math. 41, 287–307 (1983)
Iima M., Yanagita T.: Is a two-dimensional butterfly able to fly by symmetric flapping?. J. Phys. Soc. Japan 70, 5–8 (2001)
Katsurada M., Okamoto H.: A mathematical study of the charge simulation method I. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35, 507–518 (1988)
Katsurada M., Okamoto H.: The collocation points of the fundamental solution method for the potential problem. Comput. Math. Appl. 31, 123–137 (1996)
Kitagawa T.: On the numerical stability of the method of fundamental solution applied to the Dirichlet problem. Japan J. Appl. Math. 5, 123–133 (1988)
Koebe P.: Abhandlungen zur Theorie der konformen Abbildung IV, Abbildung mehrfach zusammenhängender schlichter Bereiche auf Schlitzbereiche. Acta Math. 41, 305–344 (1916)
Kythe P.K.: Computational Conformal Mapping. Birkhäuser, Boston (1998)
Lentink D., Dickson W.B., van Leeuwen J.L., Dickinson M.H.: Leading-edge vortices elevate lift of autorotating plant seeds. Science 324, 1438–1440 (2009)
Milne-Thomson L.M.: Theoretical Hydrodynamics. Dover, New York (1996)
Murashima S., Kuhara H.: An approximate method to solve two-dimensional Laplace’s equation by means of superposition of Green’s functions on a Riemann surface. J. Inform. Process. 3, 127–139 (1980)
Murota K.: Comparison of conventional and “invariant” schemes of fundamental solutions method for annular domains. Japan J. Indust. Appl. Math. 12, 61–85 (1995)
Nehari Z.: Conformal Mapping. McGraw-Hill, New York (1952)
Ogata H., Okano D., Amano K.: Numerical conformal mapping of periodic structure domains. Japan J. Indust. Appl. Math. 19, 257–275 (2002)
Ogata H., Okano D., Sugihara M., Amano K.: Unique solvability of the linear system appearing in the invariant scheme of the charge simulation method. Japan J. Indust. Appl. Math. 20, 17–35 (2003)
Ogata H., Amano K., Sugihara M., Okano D.: A fundamental solution method for viscous flow problems with obstacles in a periodic array. J. Comput. Appl. Math. 152, 411–425 (2003)
Ogata H.: A fundamental solution method for three-dimensional Stokes flow problems with obstacles in a planar periodic array. J. Comput. Appl. Math. 189, 622–634 (2006)
Ogata H., Amano K.: A fundamental solution method for three-dimensional viscous flow problems with obstacles in a periodic array. J. Comput. Appl. Math. 193, 302–318 (2006)
Ogata H., Amano K.: A fundamental solution method for two-dimensional Stokes flow problems with one-dimensional periodicity. Japan J. Indus. Appl. Math. 27, 191–215 (2010)
Okano D., Ogata H., Amano K., Sugihara M.: Numerical conformal mappings of bounded multiply connected domains by the charge simulation method. J. Comput. Appl. Math. 159, 109–117 (2003)
Sakajo T.: Equation of motion for point vortices in multiply connected circular domains. Proc. R. Soc. Lond. Ser. A Math. Phy. Eng. Sci. 465, 2589–2611 (2009)
Shiba M.: On the Riemann–Roch theorem on open Riemann surfaces. J. Math. Kyoto Univ. 11, 495–525 (1971)
Singer H., Steinbigler H., Weiss P.: A charge simulation method for the calculations of high voltage fields. IEEE Trans. Power Apparat. Syst. PAS-93, 1660–1668 (1974)
Symm G.T.: An integral equation method in conformal mapping. Numer. Math. 9, 250–258 (1966)
Symm G.T.: Numerical mapping of exterior domains. Numer. Math. 10, 437–445 (1967)
Symm G.T.: Conformal mapping of doubly-connected domains. Numer. Math. 13, 448–457 (1969)
Trefethen, L.N. (ed.) Numerical Conformal Mapping. North-Holland, Amsterdam (1986)
Acknowledgments
The authors wish to thank Emeritus Prof. M. Shiba (Hiroshima University) and Prof. T. Sakajo (Hokkaido University) for their helpful discussion.
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This work was supported by Grant-in-Aid for Scientific Research (B) 19340024, Japan Society for the Promotion of Science.
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Amano, K., Okano, D., Ogata, H. et al. Numerical conformal mappings onto the linear slit domain. Japan J. Indust. Appl. Math. 29, 165–186 (2012). https://doi.org/10.1007/s13160-012-0058-0
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DOI: https://doi.org/10.1007/s13160-012-0058-0