Journal of Turbulence Jot Methods for Computing Singular and Nearly Singular Integrals *

Many scientific problems are formulated in terms of singular integrals. We describe a simple method for computing such integrals. Our approach is to replace a singularity, or near singularity, with a regularized version, compute a sum in a standard way, and then add correction terms, which are found by asymptotic analysis near the singularity. We have used this approach for a single-layer potential on a doubly periodic surface, evaluated at a grid point on the surface. The quadrature rule so developed was used to design a convergent boundary integral method for three-dimensional water waves. In related work we have developed a method for computing a double-layer potential on a curve, evaluated at a point near the curve. Thus values of harmonic functions, with prescribed boundary conditions on a curve, can be calculated at grid points inside or outside, with only slightly extra effort for those points near the boundary. This procedure may be useful for computing fluid flow with moving boundaries. PACS number: 02.70.Pt We describe a simple and efficient approach to computing singular and nearly singular integrals. We emphasize two cases. The first is a single-layer potential on a surface in 3D, evaluated at a grid point in a coordinate system on the surface. This approach has been used to design a convergent boundary integral method for time-dependent, three-dimensional water waves [1]. The second case is a layer potential on a curve in the plane, evaluated at a point near the curve but not on it. When a problem is solved in the boundary integral or boundary element formulation, it is a separate task to find required field quantities away from the boundary. For points near the boundary, the integrals are nearly singular, that is, the integrands have large derivatives. The need for methods to compute such integrals has often been noted. In the work [2] with Ming-Chi Lai we developed a method using the approach described here. As an application, we compute harmonic functions with conditions prescribed on a general smooth ∗ This article was chosen from the Selected Proceedings of the 4th International Workshop on Vortex Flows and Related Numerical Methods (UC Santa-Barbara, 17–20 March 2002), ed E Meiburg, G H Cottet, A Ghoniem and G Koumoutsakos. c ©2002 IOP Publishing Ltd PII: S1468-5248(02)54701-9 1468-5248/02/000001+4$30.00 Nearly singular integrals boundary; an integral formulation can be used to compute values at points which are near the boundary. This method could be applied to computations of fluid flow with moving boundaries. For example, the pressure change due to a force on a boundary can be written in terms of singleand double-layer potentials. Our strategy is to replace the singularity, or near singularity, with a regularized version, compute a trapezoidal sum, which may be inaccurate, and then add correction terms. The regularization corresponds to the ‘blobs’ in vortex methods, which are used to regularize the singular integral giving the velocity from the vorticity. The correction terms are found by asymptotic analysis near the singularity. They account for the discretization error as well as the error introduced by smoothing. Because the corrections are local, the full computation could be done efficiently using a fast summation method. For a smooth function, integrated without boundaries, the trapezoidal rule on a regular grid is high-order accurate. For integrands with singularities such as those in the fundamental solution of the Laplacian, the situation is very different. The error is not high order, but there is a regular expansion which can be exploited. To illustrate this, we compare