An analysis of the steepest descent method to efficiently compute the three-dimensional acoustic single-layer operator in the high-frequency regime

Using the Cauchy integral theorem, we develop the application of the steepest descent method to efficiently compute the three-dimensional acoustic single-layer integral operator for large wave numbers. Explicit formulas for the splitting points are derived in the one-dimensional case to build suitable complex integration paths. The construction of admissible steepest descent paths is next investigated and some of their properties are stated. Based on these theoretical results, we derive the quadrature scheme of the oscillatory integrals first in dimension one and then extend the methodology to three-dimensional planar triangles. Numerical simulations are finally reported to illustrate the accuracy and efficiency of the proposed approach.

Automated summary

In this paper, the Cauchy integral theorem is utilized to develop the steepest descent method for efficiently computing the three-dimensional acoustic single-layer integral operator for large wave numbers. The explicit formulas for the splitting points are derived and the construction of admissible steepest descent paths is investigated. Based on the theoretical results, the quadrature scheme of the oscillatory integrals is derived in one dimension and extended to three-dimensional planar triangles. Numerical simulations are conducted to demonstrate the accuracy and efficiency of the proposed approach.