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An Adjoint-Based Approach for a Class of Nonlinear Fokker-Planck Equations and Related Systems

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PDE Models for Multi-Agent Phenomena

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Abstract

Here, we introduce a numerical approach for a class of Fokker-Planck (FP) equations. These equations are the adjoint of the linearization of Hamilton-Jacobi (HJ) equations. Using this structure, we show how to transfer properties of schemes for HJ equations to FP equations. Hence, we get numerical schemes with desirable features such as positivity and mass-preservation. We illustrate this approach in examples that include mean-field games and a crowd motion model.

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Acknowledgements

The author “D. Gomes” was partially supported by KAUST baseline and start-up funds and by KAUST OSR-CRG2017-3452. The author “A. Festa” was partially supported by the Haute-Normandie Regional Council via the M2NUM project.

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Correspondence to Diogo A. Gomes .

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Festa, A., Gomes, D.A., Velho, R.M. (2018). An Adjoint-Based Approach for a Class of Nonlinear Fokker-Planck Equations and Related Systems. In: Cardaliaguet, P., Porretta, A., Salvarani, F. (eds) PDE Models for Multi-Agent Phenomena. Springer INdAM Series, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-030-01947-1_4

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