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Parsimonious convolution quadrature

by J. M. Melenk and J. Nick

(Report number 2024-34)

Abstract
We present a method to rapidly approximate convolution quadrature (CQ) approximations, based on a piecewise polynomial interpolation of the Laplace domain operator, which we call the \emph{parsimonious} convolution quadrature method. For implicit Euler and second order backward difference formula based discretizations, we require \(O(\sqrt{N}\log N)\) evaluations in the Laplace domain to approximate \(N\) time steps of the convolution quadrature method to satisfactory accuracy. The methodology proposed here differentiates from the well-understood fast and oblivious convolution quadrature \cite{SLL06}, since it is applicable to Laplace domain operator families that are only defined and polynomially bounded on a positive half space, which includes acoustic and electromagnetic wave scattering problems. The methods is applicable to linear and nonlinear integral equations. To elucidate the core idea, we give a complete and extensive analysis of the simplest case and derive worst-case estimates for the performance of parsimonious CQ based on the implicit Euler method. For sectorial Laplace transforms, we obtain methods that require \(O(\log^2 N)\) Laplace domain evaluations on the complex right-half space. We present different implementation strategies, which only differ slightly from the classical realization of CQ methods. Numerical experiments demonstrate the use of the method with a time-dependent acoustic scattering problem, which was discretized by the boundary element method in space.

Keywords: convolution quadrature, wave scattering, time-dependent boundary integral equations, boundary element method

@Techreport{MN24_1116,
  author = {J. M. Melenk and J. Nick},
  title = {Parsimonious convolution quadrature},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-34},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-34.pdf },
  year = {2024}
}

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