This paper presents a method to evaluate the near-singular line integrals that solve elliptic boundary value problems in planar and axisymmetric geometries. The integrals are near-singular for target points not on, but near the boundary, and standard quadratures lose accuracy as the distance d to the boundary decreases. The method is based on Taylor series approximations of the integrands that capture the near-singular behaviour and can be integrated in closed form. It amounts to applying the trapezoid rule with meshsize h, and adding a correction for each of the basis functions in the Taylor series. The corrections are computed at a cost of \(O(n_w)\) per target point, where typically, \(n_w\)=10–40. Any desired order of accuracy can be achieved using the appropriate number of terms in the Taylor series expansions. Two explicit versions of order \(O(h^2)\) and \(O(h^3)\) are listed, with errors that decrease as \(d\rightarrow 0\). The method is applied to compute planar potential flow past a plate and past two cylinders, as well as long-time vortex sheet separation in flow past an inclined plate. These flows illustrate the significant difficulties introduced by inaccurate evaluation of the near-singular integrals and their resolution by the proposed method. The corrected results converge at the analytically predicted rates.

本文提出了一种评估近奇异线积分的方法，该积分可解决平面和轴对称几何中的椭圆边值问题。对于不在边界上而是在边界附近的目标点，积分是近奇异的，并且随着到边界的距离d减小，标准积分失去精度。该方法基于捕获近奇异行为的被积函数的泰勒级数近似，并且可以以封闭形式积分。它相当于应用网格大小为h的梯形规则，并为泰勒级数中的每个基函数添加校正。以每个目标点\(O(n_w)\)的成本计算修正，其中通常\(n_w\)=10–40。在泰勒级数展开式中使用适当数量的项可以实现任何所需的精度顺序。列出了订单\(O(h^2)\)和\(O(h^3)\) 的两个显式版本，误差随着\(d\rightarrow 0\) 减少。该方法用于计算通过一个板和通过两个圆柱体的平面势流，以及通过倾斜板的流动中的长时间涡片分离。这些流程说明了由所提出的方法对近奇异积分及其分辨率的不准确评估所带来的重大困难。校正后的结果收敛于分析预测的速率。