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Mathematics of Computation

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A second-order numerical method for the aggregation equations


Authors: José A. Carrillo, Ulrik S. Fjordholm and Susanne Solem
Journal: Math. Comp. 90 (2021), 103-139
MSC (2010): Primary 35R09, 35D30, 35Q92, 65M12, 65M08
DOI: https://doi.org/10.1090/mcom/3563
Published electronically: August 18, 2020
MathSciNet review: 4166455
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Abstract: Inspired by so-called TVD limiter-based second-order schemes for hyperbolic conservation laws, we develop a formally second-order accurate numerical method for multi-dimensional aggregation equations. The method allows for simulations to be continued after the first blow-up time of the solution. In the case of symmetric, $\lambda$-convex potentials with a possible Lipschitz singularity at the origin, we prove that the method converges in the Monge–Kantorovich distance towards the unique gradient flow solution. Several numerical experiments are presented to validate the second-order convergence rate and to explore the performance of the scheme.


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Additional Information

José A. Carrillo
Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
ORCID: 0000-0001-8819-4660
Email: carrillo@maths.ox.ac.uk

Ulrik S. Fjordholm
Affiliation: Department of Mathematics, University of Oslo, 0851 Oslo, Norway
MR Author ID: 887509
ORCID: 0000-0001-8528-5786
Email: ulriksf@math.uio.no

Susanne Solem
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
MR Author ID: 1106760
Email: susanne.solem@ntnu.no

Keywords: Aggregation equations, numerical methods, weak measure solutions, measure reconstruction.
Received by editor(s): November 19, 2019
Received by editor(s) in revised form: April 22, 2020
Published electronically: August 18, 2020
Additional Notes: The first author was partially supported by the EPSRC grant number EP/P031587/1 and the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363).
Article copyright: © Copyright 2020 American Mathematical Society