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Convergence of a linearly transformed particle method for aggregation equations
Martin Campos Pinto1,2, José A Carrillo3, Frédérique Charles1,2
11CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
Numerische Mathematik
|August 28, 2018
Summary
This study introduces a particle method for the aggregation equation, proving its accuracy for both smooth and singular interaction forces. The method demonstrates reliable convergence for various solution types and interaction complexities.
Area of Science:
- Numerical Analysis
- Partial Differential Equations
- Mathematical Modeling
Background:
- The aggregation equation models collective behavior phenomena.
- Efficient numerical methods are crucial for analyzing solutions.
- Understanding convergence properties is key for method validation.
Purpose of the Study:
- To analyze a linearly transformed particle method for the aggregation equation.
- To establish convergence guarantees for smooth and singular interaction forces.
- To provide error estimates for measure-valued solutions.
Main Methods:
- Application of a linearly transformed particle method.
- Derivation of convergence estimates in various mathematical norms (e.g., L2, L-infinity).
- Analysis of solutions in bounded Lipschitz distance for measure-valued cases.
Main Results:
- Convergence estimates are provided for smooth interaction forces, dependent on initial data regularity.
- Convergence in bounded Lipschitz distance is shown for measure-valued solutions.
- Error convergence in L-infinity norm is established for singular interaction forces up to the solution's existence time.
Conclusions:
- The linearly transformed particle method is a robust approach for the aggregation equation.
- The method offers reliable convergence properties for diverse interaction types and solution regularities.
- This work contributes to the numerical analysis of collective behavior models.
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