I'm a research associate at the University of Oxford, associated with the Oxford Centre for Nonlinear Partial Differential Equations at the Mathematical Institute.

Previously, I was a postdoctoral researcher at the Université de Lille, in the ANEDP and Inria RAPSODI groups, where I worked with Thomas Rey. I earned my doctorate at Imperial College London, advised by José Antonio Carrillo and Pierre Degond.

My main interest is the numerical analysis of kinetic equations and other PDEs. I'm also interested in collective dynamics, self-organisation, and control of agent-based models.

Rafael Bailo

Thomas Rey

Rafael Bailo · Thomas Rey

**Journal of Computational Physics**, 458, 2022.

```
@Article{BR2022
title={Projective and telescopic projective integration for non-linear kinetic mixtures},
author={Bailo, Rafael and Rey, Thomas},
journal={J Comput Phys},
year={2022},
doi={10.1016/j.jcp.2022.111082},
volume={458},
archivePrefix={arXiv},
arxivId={2106.08811},
eprint={2106.08811},
}
```

We propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and Bhatnagar-Gross-Krook (BGK) equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the eﬃcient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We validate the schemes on a range of scenarios, demonstrating its prowess in dealing with extreme mass ratios, ﬂuid instabilities, and other complex phenomena.

We propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and Bhatnagar-Gross-Krook (BGK) equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the eﬃcient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We validate the schemes on a range of scenarios, demonstrating its prowess in dealing with extreme mass ratios, ﬂuid instabilities, and other complex phenomena.

Rafael Bailo

José Antonio Carrillo

Jingwei Hu

Rafael Bailo · José Antonio Carrillo · Jingwei Hu

**arXiv: 2110.08186**, 2021.

```
@Article{BCH2021
title={Bound-preserving finite-volume schemes for systems of continuity equations with saturation},
author={Bailo, Rafael and Carrillo, José Antonio and Hu, Jingwei},
journal={arXiv: 2110.08186},
year={2021},
doi={10.48550/arXiv.2110.08186},
archivePrefix={arXiv},
arxivId={2110.08186},
eprint={2110.08186},
}
```

We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever applicable. These results are illustrated through extensive numerical simulations which explore known behaviours in biology and showcase new phenomena not yet described by the literature.

We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever applicable. These results are illustrated through extensive numerical simulations which explore known behaviours in biology and showcase new phenomena not yet described by the literature.