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Grid-free weighted particle method applied to the Vlasov–Poisson equation

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Abstract

We study a grid-free particle method based on following the evolution of the characteristics of the Vlasov–Poisson system, and we show that it converges for smooth enough initial data. This method is built as a combination of well-studied building blocks—mainly time integration and integral quadratures—and allows to obtain arbitrarily high orders. By making use of the Non-Uniform Fast Fourier Transform, the overall computational complexity is \( {\mathcal {O}}(P \log P + K^d \log K^d) \), where \( P \) is the total number of particles and where we only keep the Fourier modes \( k \in ({\mathbb {Z}}^d)^* \) such that \( k_1^2 + \dots + k_d^2 \le K^2 \). Some numerical results are given for the Vlasov–Poisson system in the one-dimensional case.

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Acknowledgements

The author would like to thank his PhD advisors, Nicolas Crouseilles and Erwan Faou, for fruitful discussions and their valuable insights. The author would also like to thank the Centre Henri Lebesgue, program ANR-11- LABX-0020-0. This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200—EUROfusion). Views and opinions expressed are however those of the author only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

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  1. IRMAR, UMR CNRS 6625, Rennes University, Rennes, France

    Yoann Le Henaff

  2. MINGuS Project Team, Inria Centre at Rennes University, Rennes, France

    Yoann Le Henaff

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Le Henaff, Y. Grid-free weighted particle method applied to the Vlasov–Poisson equation. Numer. Math. 155, 289–344 (2023). https://doi.org/10.1007/s00211-023-01378-4

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