基于梯形法则的近奇异积分奇点交换方法
Singularity swapping method for nearly singular integrals based on trapezoidal rule
AI总结：
本文提出一种基于梯形法则和三角插值的奇点交换方法，用于精确计算任意接近目标的近奇异积分，实现谱精度，并扩展到分段解析曲线，数值算例验证了高阶精度。
摘要：
近奇异积分的精确计算在许多基于边界积分方程的数值方法中起着重要作用。本文提出了一种奇点交换方法的变体，以精确评估任意接近目标的层势。我们的方法基于全局梯形法则和三角插值，从而得到一个显式求积公式。该方法在闭合解析曲线上对近奇异积分实现了谱精度。为了从复化距离函数中提取奇点，提出了一种基于轮廓积分的有效求根方法。通过变量替换，我们还将求积方法扩展到分段解析曲线上的积分。针对拉普拉斯方程和亥姆霍兹方程的数值算例表明，对于任意接近的场评估，可以实现高阶精度。
英文摘要：
Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer potentials for arbitrarily close targets. Our method is based on the global trapezoidal rule and trigonometric interpolation, resulting in an explicit quadrature formula. The method achieves spectral accuracy for nearly singular integrals on closed analytic curves. In order to extract the singularity from the complexified distance function, an efficient root finding method is proposed based on contour integration. Through the change of variables, we also extend the quadrature method to integrals on the piecewise analytic curves. Numerical examples for Laplace's and Helmholtz equations show that high order accuracy can be achieved for arbitrarily close field evaluation.
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