Quadrature by Two Expansions for Evaluating Helmholtz Layer Potentials

TL;DR

The paper introduces QB2X, a novel numerical integration method for Helmholtz layer potentials that improves accuracy and efficiency over existing QBX techniques, especially for complex geometries.

Contribution

The QB2X method explicitly incorporates boundary geometry nonlinearity into plane wave expansions, enhancing accuracy and convergence for Helmholtz layer potential evaluations.

Findings

QB2X achieves higher accuracy than QBX for complex geometries.

Numerical results confirm QB2X's superior performance and convergence.

QB2X is more suitable for efficient Helmholtz solutions with intricate boundaries.

Abstract

In this paper, a Quadrature by Two Expansions (QB2X) numerical integration technique is developed for the single and double layer potentials of the Helmholtz equation in two dimensions. The QB2X method uses both local complex Taylor expansions and plane wave type expansions to achieve a resulting representation which is numerically accurate for all target points (interior, exterior, or exactly on the boundary) inside a leaf box in the fast multipole method (FMM) hierarchical tree structure. Compared to the original Quadrature by Expansion (QBX) method, the QB2X method explicitly includes the nonlinearity from the boundary geometry in the plane wave expansions, thereby providing for higher order representations of both the boundary geometry and density functions in the integrand, with its convergence following standard FMM error analysis. Numerical results are presented to demonstrate the performance of the QB2X method for Helmholtz layer potentials and its comparison with the original QBX method for both flat and curved boundaries with various densities. The QB2X method overcomes the challenges of the original QBX method, and is better suited for efficient solutions of the Helmholtz equation with complex geometries.