I'm a research associate at the University of Oxford, affiliated with the Oxford Centre for Nonlinear Partial Differential Equations at the Mathematical Institute. In addition, I serve as a non-stipendiary lecturer at the Queen's college.

Prior to this, I worked as a postdoctoral researcher at the Université de Lille with the ANEDP and Inria RAPSODI groups, under the supervision of Thomas Rey. I earned my doctorate at Imperial College London, where my advisors were José Antonio Carrillo and Pierre Degond.

My primary area of interest lies in the numerical analysis of kinetic equations and other partial differential equations (PDEs). I'm also fascinated by collective dynamics, self-organisation, and the control of agent-based models.

Rafael Bailo · José Antonio Carrillo · Serafim Kalliadasis · Sergio P. Perez

**arXiv: 2105.05351**, 2023.

```
@Article{BCK2023
title={Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn-Hilliard equation},
author={Bailo, Rafael and Carrillo, José Antonio and Kalliadasis, Serafim and Perez, Sergio P.},
journal={arXiv: 2105.05351},
year={2023},
doi={10.48550/arXiv.2105.05351},
archivePrefix={arXiv},
arxivId={2105.05351},
eprint={2105.05351},
}
```

We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy. Our numerical framework is applicable to a variety of free-energy potentials, including Ginzburg-Landau and Flory-Huggins, to general wetting boundary conditions, and to degenerate mobilities. Its central thrust is the upwind methodology, which we combine with a semi-implicit formulation for the free-energy terms based on the classical convex-splitting approach. The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature, which allows to efficiently solve higher-dimensional problems with a simple parallelisation. The numerical schemes are validated and tested through a variety of examples, in different dimensions, and with various contact angles between droplets and substrates.

We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy. Our numerical framework is applicable to a variety of free-energy potentials, including Ginzburg-Landau and Flory-Huggins, to general wetting boundary conditions, and to degenerate mobilities. Its central thrust is the upwind methodology, which we combine with a semi-implicit formulation for the free-energy terms based on the classical convex-splitting approach. The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature, which allows to efficiently solve higher-dimensional problems with a simple parallelisation. The numerical schemes are validated and tested through a variety of examples, in different dimensions, and with various contact angles between droplets and substrates.

Rafael Bailo

José Antonio Carrillo

Jingwei Hu

Rafael Bailo · José Antonio Carrillo · Jingwei Hu

**SIAM Journal on Applied Mathematics**, 2023 (to appear).

```
@Article{BCH2023
title={Bound-preserving finite-volume schemes for systems of continuity equations with saturation},
author={Bailo, Rafael and Carrillo, José Antonio and Hu, Jingwei},
journal={SIAM J Appl Math (to appear)},
year={2023},
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.