Methods for numerical conformal mapping
Nonlinear integral equations for the boundary functions that determine conformal transformations in two dimensions are developed and analyzed. One of these equations has a nonsingular logarithmic kernel and is especially well suited for numerical computations of conformal maps including those that deal with regions having highly distorted boundaries. Numerical procedures based on interspersed Gaussian quadrature for approximating the integrals and a Newton--Raphson technique to solve the resulting nonlinear algebraic equations are described. The Newton--Raphson iteration converges reliably with very crude initial approximations. Numerical examples are given for the mapping of a half-infinite region with periodic boundary onto a half-plane, with up to 9-figure accuracy for values of the map function on the boundary and for its first derivatives. The examples include regions bounded by spike curves characteristic of Rayleigh--Taylor instability phenomena. A differential equation is derived that relates changes in the map function to changes of the boundary. This is relevant to potential problems for regions with time-dependent boundaries. Further nonsingular integral formulas are derived for conformal mapping in a variety of geometries and for application to the boundary-value problems of potential theory. 4 figures, 4 tables.
- Research Organization:
- Los Alamos Scientific Lab., NM (USA)
- DOE Contract Number:
- W-7405-ENG-36
- OSTI ID:
- 5969240
- Report Number(s):
- LA-7836-MS
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
658000* -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
BOUNDARY-VALUE PROBLEMS
CONFORMAL MAPPING
DATA
DATA FORMS
DIFFERENTIAL EQUATIONS
EQUATIONS
FLUID FLOW
FLUID MECHANICS
INCOMPRESSIBLE FLOW
INFORMATION
INSTABILITY
INTEGRAL EQUATIONS
ISOLATED VALUES
ITERATIVE METHODS
MECHANICS
NEWTON METHOD
NONLINEAR PROBLEMS
NUMERICAL DATA
NUMERICAL SOLUTION
POTENTIALS
QUADRATURES
RAYLEIGH-TAYLOR INSTABILITY
THEORETICAL DATA
TOPOLOGICAL MAPPING
TRANSFORMATIONS
TWO-DIMENSIONAL CALCULATIONS