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On the accurate evaluation of unsteady Stokes layer potentials in moving two-dimensional geometries

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Abstract

Two fundamental difficulties are encountered in the numerical evaluation of time-dependent layer potentials. One is the quadratic cost of history dependence, which has been successfully addressed by splitting the potentials into two parts—a local part that contains the most recent contributions and a history part that contains the contributions from all earlier times. The history part is smooth, easily discretized using high-order quadratures, and straightforward to compute using a variety of fast algorithms. The local part, however, involves complicated singularities in the underlying Green’s function. Existing methods, based on exchanging the order of integration in space and time, are able to achieve high-order accuracy, but are limited to the case of stationary boundaries. Here, we present a new quadrature method that leaves the order of integration unchanged, making use of a change of variables that converts the singular integrals with respect to time into smooth ones. We have also derived asymptotic formulas for the local part that lead to fast and accurate hybrid schemes, extending earlier work for scalar heat potentials and applicable to moving boundaries. The performance of the overall scheme is demonstrated via numerical examples.

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Funding

S. Jiang was supported by the National Science Foundation under grant DMS-1720405 and by the Flatiron Institute, a division of the Simons Foundation.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Leslie Greengard

  2. Flatiron Institute, Simons Foundation, New York, NY, 10010, USA

    Leslie Greengard & Jun Wang

  3. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102, USA

    Shidong Jiang

Authors
  1. Leslie Greengard
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  2. Shidong Jiang
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  3. Jun Wang
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Correspondence to Leslie Greengard.

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Communicated by: Gunnar J Martinsson

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Greengard, L., Jiang, S. & Wang, J. On the accurate evaluation of unsteady Stokes layer potentials in moving two-dimensional geometries. Adv Comput Math 46, 17 (2020). https://doi.org/10.1007/s10444-020-09760-8

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  • DOI: https://doi.org/10.1007/s10444-020-09760-8

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