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Boundary integral equations and Nystrom discretization provide a powerfultool for the solution of Laplace and Helmholtz boundary value problems.However, often a weakly-singular kernel arises, in which case specializedquadratures that modify the matrix entries near the diagonal are needed toreach a high accuracy. We describe the construction of four differentquadratures which handle logarithmically-singular kernels. Only smoothboundaries are considered, but some of the techniques extend straightforwardlyto the case of corners. Three are modifications of the global periodictrapezoid rule, due to Kapur-Rokhlin, to Alpert, and to Kress. The fourth is amodification to a quadrature based on Gauss-Legendre panels due toKolm-Rokhlin; this formulation allows adaptivity. We compare in numericalexperiments the convergence of the four schemes in various settings, includinglow- and high-frequency planar Helmholtz problems, and 3D axisymmetric Laplaceproblems. We also find striking differences in performance in an iterativesetting. We summarize the relative advantages of the schemes.

High-order accurate Nystrom discretization of integral equations with weakly singular kernels on smooth curves in the plane