Application of multigrid methods for integral equations to two problems from fluid dynamics

Abstract

Multigrid methods are applied in order to solve efficiently the nonsparse systems of equations that occur in the numerical solution of the following problems from fluid dynamics: (1) calculation of potential flow around bodies and (2) calculation of oscillating disk flow. Problem (1) is reformulated as a boundary integral equation of the second kind that is approximated by a first-order panel method resulting in a full system of equations. This method is in widespread use for aerodynamic computations. The second problem is described by the Navier-Stokes and continuity equations. By means of the von Kármán similarity transformations these equations are reduced to a nonlinear system of parabolic equations which are approximated by implicit finite difference techniques. From the periodic conditions in time one obtains a nonsparse system of equations. For these two problems from fluid dynamics the fast convergence of multigrid methods for integral equations is established by numerical experiments.