Computation of Nearly Singular Integrals with Applications to Fluid Dynamics

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0102356BealeThe proposed work is concerned with the development of improved methods for computing singular or nearly singular integrals, and with applications to numerical methods for simulating fluid flow with moving boundaries. In many scientific problems, quantities of interest can be written in terms of integrals such as a double layer potential on a boundary. When such an integral is evaluated near the boundary, usual quadrature rules are inaccurate because of large derivatives in the integrand. It is desirable to calculate such values in a way that is simple and efficient, without requiring a special choice of quadrature points. The present approach uses regularization of the singularity and a standard quadrature rule, followed by local corrections derived from asymptotic analysis. This technique will be used to design an alternate version of the immersed boundary method of C. Peskin, a computational method which has been used to model various biological processes. In this method viscous, incompressible fluid is influenced by a membrane or interface which exerts a force on the fluid. In two dimensions, the force can be expressed in terms of nearly singular integrals to which the proposed approach can be applied. It is hoped that this modification will extend the usefulness of the immersed boundary method in realistic applications. Application may also be made to methods for computing flow with moving boundaries separating different fluids. The technique for nearly singular integrals will be extended to surface integrals in three-space, generalizing work already done for marker points on surfaces. Related work will concern interfaces in inviscid, potential flow using improved regularizations of boundary integrals.Many scientific processes involve moving boundaries, for example, the boundary of a drop of one fluid moving through another, or a membrane in living tissue. Numerical models of such processes encounter special difficulties, since it is simplest to compute quantities at a set of points which are regularly spaced and unchanging. There are presently several numerical approaches which avoid changing fundamentally the way the region is approximated at each new time. In some cases important quantities can be written as integrals over boundaries. These integrals arise as solutions of differential equations and are usually singular, or nearly singular; that is, large values appear, representing the strong interaction of nearby points, as in the inverse square law of gravity. Special techniques are needed to calculate such integrals accurately and efficiently. This proposal is concerned with the development and application of methods of this type. The use of integral representations to modify the design of existing methods for computing fluid motion with moving boundaries might improve the ability of these numerical methods to make reliable predictions.