Numerical conformal mappings onto the linear slit domain

We propose a numerical method for the conformal mapping of unbounded multiply connected domains exterior to closed Jordan curves C1, . . . ,Cn onto a canonical linear slit domain, which is the entire plane with linear slits S1, . . . , Sn of angles θ1, . . . , θn arbitrarily assigned to the real axis, respectively. If θ1 = · · · = θn = 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.

[1]  M. Sugihara,et al.  A fundamental solution method for viscous flow problems with obstacles in a periodic array , 2003 .

[2]  Dieter Gaier,et al.  Integralgleichungen erster Art und konforme Abbildung , 1976 .

[3]  D. Gaier.  Das logarithmische Potential und die konforme Abbildung mehrfach zusammenhängender Gebiete , 1981 .

[4]  B. Fornberg.  A Numerical Method for Conformal Mappings , 1980 .

[5]  M. K. Kerimov,et al.  Applied and computational complex analysis. Vol. 1. Power series, integration, conformal mapping, location of zeros: Henrici P. xv + 682 pp., John Wiley and Sons, Inc., New York — London, 1974☆ , 1977 .

[6]  G. Symm.  Conformal mapping of doubly-connected domains , 1969 .

[7]  Hisashi Okamoto,et al.  The collocation points of the fundamental solution method for the potential problem , 1996 .

[8]  M. Shiba.  On the Riemann-Roch theorem on open Riemann surfaces , 1971 .

[9]  A. Elcrat,et al.  Schwarz-Christoffel mapping of multiply connected domains , 2004 .

[10]  David K. Kahaner,et al.  An Improved Method for Numerical Conformal Mapping , 1972 .

[11]  N. Papamichael,et al.  The use of splines and singular functions in an integral equation method for conformal mapping , 1981 .

[12]  Kazuo Murota,et al.  Comparison of conventional and “invariant” schemes of fundamental solutions method for annular domains , 1995 .

[13]  Is a Two-dimensional Butterfly Able to Fly by Symmetric Flapping? , 2001 .

[14]  Masashi Katsurada,et al.  A mathematical study of the charge simulation method I , 1988 .

[15]  Prem K. Kythe,et al.  Computational Conformal Mapping , 1998 .

[16]  Darren Crowdy,et al.  Analytical solutions for uniform potential flow past multiple cylinders , 2006 .

[17]  S. Murashima,et al.  An approximate method to solve two-dimensional Laplace's equation by means of superposition of Green's functions On a Riemann surface , 1980 .

[18]  George T. Symm,et al.  An integral equation method in conformal mapping , 1966 .

[19]  Kaname Amano,et al.  Numerical conformal mappings of bounded multiply connected domains by the charge simulation method , 2003 .

[20]  Unique solvability of the linear system appearing in the invariant scheme of the charge simulation method , 2003 .

[21]  Hidenori Ogata.  A fundamental solution method for three-dimensional Stokes flow problems with obstacles in a planar periodic array , 2006 .

[22]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[23]  M. Gutknecht.  Numerical conformal mapping methods based on function conjugation , 1986 .

[24]  Lloyd N. Trefethen,et al.  Schwarz-Christoffel Mapping , 2002 .

[25]  Hidenori Ogata,et al.  Fundamental solution method for two-dimensional Stokes flow problems with one-dimensional periodicity , 2010 .

[26]  Thomas K. DeLillo,et al.  Numerical conformal mapping of multiply connected regions by Fornberg-like methods , 1999, Numerische Mathematik.

[27]  Hidenori Ogata,et al.  A fundamental solution method for three-dimensional viscous flow problems with obstacles in a periodic array , 2006 .

[28]  Paul Koebe,et al.  Abhandlungen zur Theorie der konformen Abbildung , 1916 .

[29]  N. Papamichael,et al.  An integral equation method for the numerical conformal mapping of interior, exterior and doubly-connected domains , 1983 .

[30]  G. Symm.  Numerical mapping of exterior domains , 1967 .

[31]  Kaname Amano,et al.  A circular and radial slit mapping of unbounded multiply connected domains , 2010, JSIAM Lett..

[32]  Bengt Fornberg,et al.  A Numerical Method for Conformal Mapping of Doubly Connected Regions , 1984 .

[33]  Tobin A. Driscoll,et al.  Radial and circular slit maps of unbounded multiply connected circle domains , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[34]  Thomas K. DeLillo,et al.  A simplified Fornberg-like method for the conformal mapping of multiply connected regions-Comparisons and crowding , 2007 .

[35]  L. M. Milne-Thomson,et al.  Theoretical hydrodynamics / by L.M. Milne-Thomson , 1955 .

[36]  Kaname Amano,et al.  A Charge Simulation Method for Numerical Conformal Mapping onto Circular and Radial Slit Domains , 1998, SIAM J. Sci. Comput..

[37]  M H Dickinson,et al.  Leading-Edge Vortices Elevate Lift of Autorotating Plant Seeds , 2009, Science.

[38]  L. Trefethen.  Numerical conformal mapping , 1986 .

[39]  Kaname Amano,et al.  A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains , 1994 .

[40]  Takashi Sakajo,et al.  Equation of motion for point vortices in multiply connected circular domains , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[41]  H. Singer,et al.  A Charge Simulation Method for the Calculation of High Voltage Fields , 1974 .

[42]  T. Kitagawa,et al.  On the numerical stability of the method of fundamental solution applied to the Dirichlet problem , 1988 .

[43]  Darren Crowdy,et al.  Conformal Mappings between Canonical Multiply Connected Domains , 2006 .

[44]  Dieter Gaier,et al.  Konstruktive Methoden der konformen Abbildung , 1964 .