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<p>In this paper, we present an asymptotic analysis and a cubature method for evaluating a bivariate oscillatory integral with a stationary line. The integral is expanded into an asymptotic series in terms of negative powers of the oscillatory parameter. We derive the complete asymptotic expansion, which includes all partial derivatives at the vertices and integrals of the non-oscillatory function. Using a bivariate interpolation approach, we construct a cubature method that is effective for both small and large oscillatory parameters, unlike the traditional asymptotic expansion method. The asymptotic error of the cubature method decays rapidly, achieving arbitrarily high asymptotic orders. Numerical results are provided to demonstrate the accuracy and asymptotic order of the proposed method.</p>

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Computing the oscillatory integrals with stationary line

  • 期刊论文

  • 发表日期：

    2025年4月9日

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  • Jing Gao,
  • Mingfei Zhang,
  • Yaolin Jiang

摘要

In this paper, we present an asymptotic analysis and a cubature method for evaluating a bivariate oscillatory integral with a stationary line. The integral is expanded into an asymptotic series in terms of negative powers of the oscillatory parameter. We derive the complete asymptotic expansion, which includes all partial derivatives at the vertices and integrals of the non-oscillatory function. Using a bivariate interpolation approach, we construct a cubature method that is effective for both small and large oscillatory parameters, unlike the traditional asymptotic expansion method. The asymptotic error of the cubature method decays rapidly, achieving arbitrarily high asymptotic orders. Numerical results are provided to demonstrate the accuracy and asymptotic order of the proposed method.