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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
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

Ulrik S. Fjordholm
Affiliation: Department of Mathematics, University of Oslo, 0851 Oslo, Norway
MR Author ID: 887509
ORCID: 0000-0001-8528-5786

Susanne Solem
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
MR Author ID: 1106760

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