Quadrature by Matched Asymptotic Expansion (QBMAX): Accelerating Evaluation of Layer Potentials with Boundary Layers
Abstract
Evaluating layer potentials with rapidly decaying kernels becomes computationally challenging in the presence of boundary layers, as occurs for Helmholtz layer potentials with large imaginary wavenumbers. We introduce Quadrature by Matched Asymptotic Expansion (QBMAX), a high-order method that extends the Quadrature by Expansion (QBX) framework by incorporating the kernel's asymptotic decay behavior directly into the Taylor expansion of the potential. Numerical results in both 2D and 3D show improvements of up to four digits over QBX for the tested large-parameter cases, while the methods can perform similarly for smaller parameters or more strongly curved geometries. A symbolic weighted operation-count model indicates comparable costs for forming the two expansion kernels. The Taylor expansions have the same formal order, while QBMAX reduces the truncation-error constants in the flat-boundary model and generally exhibits less convergence degradation in the reported experiments.
Related
- Time-asymptotic expansion and fluid approximation of the 1D Boltzmann equation with hard potentials
- AMPERe: an exactly divergence-free quadrature-by-expansion evaluator for near-surface Biot--Savart and virtual-casing fields in fusion geometries
- Numerical evaluation of one-loop integrals using quadrature rules
- Asymptotic expansion of smooth functions in deterministic and iid Haar unitary matrices, and application to tensor products of matrices
- Performance Evaluation of an Adaptive Quadrature and a Double Exponential Formula Using Arbitrary-Precision Floating-Point Arithmetic
BibTeX
@misc{lin2026quadrature,
title = {{Quadrature by Matched Asymptotic Expansion (QBMAX): Accelerating Evaluation of Layer Potentials with Boundary Layers}},
author = {Chaoqi Lin and Xiaoyu Wei and Andreas Klöckner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10857},
eprint = {2609.10857},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10857},
}
RIS
TY - UNPB
TI - Quadrature by Matched Asymptotic Expansion (QBMAX): Accelerating Evaluation of Layer Potentials with Boundary Layers
AU - Chaoqi Lin
AU - Xiaoyu Wei
AU - Andreas Klöckner
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.10857
UR - https://arxiv.org/abs/2609.10857
AB - Evaluating layer potentials with rapidly decaying kernels becomes computationally challenging in the presence of boundary layers, as occurs for Helmholtz layer potentials with large imaginary wavenumbers. We introduce Quadrature by Matched Asymptotic Expansion (QBMAX), a high-order method that extends the Quadrature by Expansion (QBX) framework by incorporating the kernel's asymptotic decay behavior directly into the Taylor expansion of the potential. Numerical results in both 2D and 3D show improvements of up to four digits over QBX for the tested large-parameter cases, while the methods can perform similarly for smaller parameters or more strongly curved geometries. A symbolic weighted operation-count model indicates comparable costs for forming the two expansion kernels. The Taylor expansions have the same formal order, while QBMAX reduces the truncation-error constants in the flat-boundary model and generally exhibits less convergence degradation in the reported experiments.
ER -