Computing highly oscillatory physical optics integral on the polygonal domain by an efficient numerical steepest descent path method

In this work, the computation of physical optics (PO) type integral with the integrand of quadratic phase and amplitude is studied. First, we apply the numerical steepest descent path (NSDP) method to calculate the highly oscillatory PO integral on the triangular patch. Then, we rigorously extend the proposed NSDP method to analyze the PO integral on polygonal domains. Furthermore, the contributions of critical points on polygonal domains, including the stationary phase point, resonance and vertex points, are comprehensively studied in terms of the NSDP method. Compared to the traditional high frequency asymptotic (HFA) method, when the wave frequency is not very high but in the high frequency regime, the NSDP method has improved the PO integral accuracy by one to two digits. Meanwhile, the computational cost by using the proposed NSDP method is independent of the wave frequency.