Convergence And Numerical Simulation Of The Charge Simulation Method For Numerical Conformal Mapping

Conformal mapping,namely conformal transformation,is one of an important is-sue in complex function.It is widely used in many fields of physical problems,such as electromagnetics,optics,fluid mechanics,etc.The numerical conformal mapping method has many tremendous advantages in dealing with practical problems in en-gineering such as convenient application and high precision.The charge simulation method employees the potentials of the charge points which are outside of the problem domain to solve the approximate solution of the boundary value problem of the two-dimensional Laplace equation.The approximate solution obtained by this method has high precision for the Dirichlet boundary value problem.Since the 1980s,Japanese Amano and other mathematical scholars have done a lot of research on the numerical conformal mapping and the charge simulation method.The charge simulation method for numerical conformal mapping is proposed by A-mano,also called Amano method.The advantage of this method is that the conformal mapping approximate function has simple form and high calculation precision,there-fore the computational complexity is reduced.Since the charge simulation method for numerical conformal mapping avoids the numerical integration,the method has the merits of high calculation efficiency,and higher calculation accuracy.The maximal principle of Laplace equation is used to evaluate the error on the boundary,so the error evaluation is more precise.In this paper,the charge simulation method for numerical conformal mapping of exterior domain is primarily studied,the main work includes:(1)The basic theories of conformal mapping and the charge simulation method are briefly introduced.(2)It is proved that the approximate solution of the Dirichlet boundary value problem of the two-dimensional Laplace equation is proved to be exponential convergence. Then the convergence theorem of the charge simulation method for numerical con-formal mapping of exterior domain is theoretically proved,and it is found that the boundary error decreases exponentially.Finally,the reliability of our proposed con-clusion is verified by comparing with numerical simulation.(3)The constraint equations of numerical conformal mapping of exterior domain are constructed by the charge simulation method,which coefficient matrix is large,non-singular and sparse.Then,in order to obtain simulated the charge Q_j and transform radius ?,the WGMRES(m)method is used to solve the constrained equations,wh-ich is derived by improving the GMRES(m)method through the weighted Arnoldi process.Thus,the conformal mapping approximate function is constructed.Finally, numerical simulations are conducted to compare the proposed method with Amano method.The results show that the proposed method can construct a more accurate conformal mapping approximate function,especially when the boundary curve is not differentiable.Its error accuracy is significantly higher than that of the Amano method,which verifies the reliability and effectiveness of the proposed method.(4)The placement locations of charge simulation points and collection points of super oval curve and closed domain bounded by two symmetric Parabolas and the Cardi-oid diagram as the boundary in the numerical conformal mapping of exterior doma-in are given.