Very high-order finite volume schemes with curved domains

Finite volume methods of third or higher order require a specific treatment of the boundary conditions when dealing with a non-polygonal domain that does not exactly fit with the mesh. We also face a similar situation with internal smooth interfaces sharing two subdomains. To address this issue, several technologies have been developed since the 90's such as the isoparamatric elements, the (ghost cells) immersed boundary and the inverse Lax-Wendroff boundary treatment among others. We propose a quick overview of the traditional methods and introduce the new Reconstruction of Off-site Data (ROD) method. Basically, the idea consists, first in definitively distinguishing the computational domain (cells or nodes where the solution is computed) to the physical one and, secondly, in \"transporting\" the boundary conditions prescribed on the real boundary to the computing domain. To this end, specific local polynomial reconstructions that contains a fingerprint of the boundary conditions are proposed and used to schemes that achieve up to sixth-order of accuracy. Several applications will be proposed in the context of the finite volume (flux reconstruction) and finite difference (ghost cells) for the convection diffusion equation and the Euler system.