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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A blob method for the aggregation equation
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by Katy Craig and Andrea L. Bertozzi PDF
Math. Comp. 85 (2016), 1681-1717

Abstract:

Motivated by classical vortex blob methods for the Euler equations, we develop a numerical blob method for the aggregation equation. This provides a counterpoint to existing literature on particle methods. By regularizing the velocity field with a mollifier or “blob function”, the blob method has a faster rate of convergence and allows a wider range of admissible kernels. In fact, we prove arbitrarily high polynomial rates of convergence to classical solutions, depending on the choice of mollifier. The blob method conserves mass and the corresponding particle system is energy decreasing for a regularized free energy functional and preserves the Wasserstein gradient flow structure. We consider numerical examples that validate our predicted rate of convergence and illustrate qualitative properties of the method.
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Additional Information
  • Katy Craig
  • Affiliation: Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, California 90095-1555
  • Email: kcraig@math.ucla.edu
  • Andrea L. Bertozzi
  • Affiliation: Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, California 90095-1555
  • MR Author ID: 265966
  • Email: bertozzi@math.ucla.edu
  • Received by editor(s): May 29, 2014
  • Received by editor(s) in revised form: December 13, 2014, and January 7, 2015
  • Published electronically: December 4, 2015
  • Additional Notes: This work was supported by NSF grants CMMI-1435709, DMS-0907931, DMS-1401867, and EFRI-1024765, as well as NSF grant 0932078 000, which supported Craig’s visit and Bertozzi’s residence at the Mathematical Sciences Research Institute during Fall 2013.
  • © Copyright 2015 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
  • Journal: Math. Comp. 85 (2016), 1681-1717
  • MSC (2010): Primary 35Q35, 35Q82, 65M15, 82C22
  • DOI: https://doi.org/10.1090/mcom3033
  • MathSciNet review: 3471104