Quadrature formulas for periodic functions and their application to the boundary element method. (Russian, English) Zbl 1199.65390

Petrov, A. G.

Zh. Vychisl. Mat. Mat. Fiz. 48, No. 8, 1344-1361 (2008); translation in Comput. Math. Math. Phys. 48, No. 8, 1266-1283 (2008).

Summary: Two-dimensional and axisymmetric boundary value problems for the Laplace equation in a domain bounded by a closed smooth contour are considered. The problems are reduced to integral equations with a periodic singular kernel, where the period is equal to the length of the contour. Taking into account the periodicity property, high-order accurate quadrature formulas are applied to the integral operator. As a result, the integral equations are reduced to a system of linear algebraic equations. This substantially simplifies the numerical schemes for solving boundary value problems and considerably improves the accuracy of approximation of the integral operator. The boundaries are specified by analytic functions, and the remainder of the quadrature formulas decreases faster than any power of the integration step size. The examples include the two-dimensional potential inviscid circulation flow past a single blade or a grid of blades; the axisymmetric flow past a torus; and free-surface flow problems, such as wave breakdown, standing waves, and the development of Rayleigh-Taylor instability.

MSC:

65N38	Boundary element methods for boundary value problems involving PDEs
65D32	Numerical quadrature and cubature formulas
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
76B07	Free-surface potential flows for incompressible inviscid fluids
76E09	Stability and instability of nonparallel flows in hydrodynamic stability
76M15	Boundary element methods applied to problems in fluid mechanics

Keywords:

Laplace equation; boundary value problems; integral equation method; quadrature formulas; numerical examples; periodic singular kernel; two-dimensional potential inviscid circulation flow; axisymmetric flow; free-surface flow; standing waves; Rayleigh-Taylor instability