Computational Methods for Singular and Nearly Singular Integrals with Applications to Fluid Dynamics

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The purpose of this work is to develop efficient methods for computing singular or nearly singular integrals and to apply the methods to the numerical simulation of fluid flow with moving boundaries. The mathematical formulation of scientific problems often involves singular integrals, such as a harmonic potential function due to a layer of sources on a curve or surface. For points near the source, evaluation is not routine because of large derivatives. The approach of this work, developed in earlier NSF-funded research, is to regularize the integral in a systematic way, evaluate at grid points as for a standard integral, and then add local correction terms. The corrections are derived analytically. For surfaces, overlapping grids are used. The work will be in several parts: A class of static problems for 3D potentials defined through surface integrals will be treated, extending earlier work. In order to allow for boundaries that are not smooth, a method will be derived for computing integrals on curves with corners in 2D. Application to the computation of moving boundaries in viscous, incompressible 2D fluid flow will be tested as a possible improvement in the immersed boundary method or immersed interface method. Computations will be done for a moving boundary or interface between two different fluids in 3D without viscosity; water waves are one important case, and in other cases regularization will be used to control physical instabilities.Scientific processes often involve moving boundaries, such as a drop of one fluid moving through another, or the motion of an elastic membrane in living tissue. Numerical modeling of such phenomena involves special difficulties, and several approaches are in use. Important quantities, such as a jump in pressure, can often be written as singular integrals like the ones described. The techniques developed in this work could be used to incorporate integral calculations into numerical methods for viscous fluid flow with moving boundaries. If this leads to improvement in these methods, they could be more widely useful for predictions of two-fluid systems or biological processes with moving membranes. The application to flow without viscosity could improve understanding of fully nonlinear water waves and the onset of mixing in an unstable fluid layer.