Quadrature Formulas for Periodic Functions and Their Application to the Boundary Element Method

摘要

Two-dimensional and axisymmetric boundary value problems for the Laplace equation in adomain bounded by a closed smooth contour are considered. The problems are reduced to integral equa-tions with a periodic singular kernel, where the period is equal to the length of the contour. Taking intoaccount the periodicity property, high-order accurate quadrature formulas are applied to the integraloperator. As a result, the integral equations are reduced to a system of linear algebraic equations. Thissubstantially simplifies the numerical schemes for solving boundary value problems and considerablyimproves the accuracy of approximation of the integral operator. The boundaries are specified by ana-lytic functions, and the remainder of the quadrature formulas decreases faster than any power of the inte-gration step size. The examples include the two-dimensional potential inviscid circulation flow past asingle blade or a grid of blades; the axisymmetric flow past a torus; and free-surface flow problems, suchas wave breakdown, standing waves, and the development of Rayleigh–Taylor instability.