Rapid methods for the conformal mapping of multiply connected regions

AbstractWe present fast methods for the conformal mapping of simply, doubly and multiply connected regions onto certain canonical regions in the plane. Our mapping procedure consists of two parts. First we solve an integral equation on the boundary of the region we wish to map. The solution of this integral equation is needed to determine the boundary correspondence. We have chosen to use the integral equation formulation of Mikhlin. Although it is not widely used, this formulation has the advantage that it leads to integral equations of the second kind with unique solutions and bounded kernels. The solutions are also periodic, allowing for effective use of the trapezoid rule. Once we have solved the integral equation we use a rapid method we have previously developed to determine the mapping function in the interior of the region. This method makes use of fast Poisson solvers, and thereby circumvents the difficulties associated with computing integrals at points near the boundary of the region, and avoids the expense of computing many integrals. We also provide results of numerical experiments

An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

We present an immersed interface method for the vorticity-velocity form of the 2D Navier Stokes equations that directly addresses challenges posed by multiply connected domains, nonconvex obstacles, and the calculation of force distributions on immersed surfaces. The immersed interface method is re-interpreted as a polynomial extrapolation of flow quantities and boundary conditions into the obstacle, reducing its computational and implementation complexity. In the flow, the vorticity transport equation is discretized using a conservative finite difference scheme and explicit Runge-Kutta time integration. The velocity reconstruction problem is transformed to a scalar Poisson equation that is discretized with conservative finite differences, and solved using an FFT-accelerated iterative algorithm. The use of conservative differencing throughout leads to exact enforcement of a discrete Kelvin's theorem, which provides the key to simulations with multiply connected domains and outflow boundaries. The method achieves second order spatial accuracy and third order temporal accuracy, and is validated on a variety of 2D flows in internal and free-space domains