## Section: New Results

### The Semi-Lagrangian method on curvilinear grids

Participants : Aurore Back, Adnane Hamiaz, Michel Mehrenberger [correspondent] , Pierre Navaro, Hocine Sellama, Eric Sonnendrücker.

We study the semi-Lagrangian method on curvilinear grids [18] , [9] . The classical backward semi-Lagrangian method preserves constant states but is not mass conservative. Natural reconstruction of the field permits nevertheless to have at least first order in time conservation of mass, even if the spatial error is large. Interpolation is performed with classical cubic splines and also cubic Hermite interpolation with arbitrary reconstruction order of the derivatives. High odd order reconstruction of the derivatives is shown to be a good ersatz of cubic splines which do not behave very well as time step tends to zero. A conservative semi-Lagrangian scheme is then described; here conservation of mass is automatically satisfied and constant states are shown to be preserved up to first order in time.

Semi-Lagrangian guiding center simulations are performed on sinusoidal perturbations of cartesian grids, and on deformed polar grids with different boundary conditions. Key ingredients are: the use of a B-spline finite element solver for the Poisson equation and the classical backward semi-Lagrangian method (BSL) for the advection. We are able to reproduce standard Kelvin-Helmholtz and diocotron instability tests on such grids. When the perturbation leads to a strong distorted mesh, we observe that the solution differs if one takes standard numerical parameters that are used in the cartesian reference case. We can recover good results together with correct mass conservation, by diminishing the time step.