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A blob method for the aggregation equation


Authors: Katy Craig and Andrea L. Bertozzi
Journal: Math. Comp. 85 (2016), 1681-1717
MSC (2010): Primary 35Q35, 35Q82, 65M15, 82C22
DOI: https://doi.org/10.1090/mcom3033
Published electronically: December 4, 2015
MathSciNet review: 3471104
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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
Email: bertozzi@math.ucla.edu

DOI: https://doi.org/10.1090/mcom3033
Keywords: Aggregation equation, vortex blob method, particle method
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.
Article copyright: © Copyright 2015 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.