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Section: New Results

Field-aligned interpolation for gyrokinetics

Participants : Yaman Güclü, Philippe Helluy, Guillaume Latu, Michel Mehrenberger, Laura Mendoza, Eric Sonnendrücker, Maurizio Ottaviani.

This work is devoted to the study of field-aligned interpolation in semi-Lagrangian codes. This work has been initiated in 2013; this year the article has been accepted [5]. In the context of numerical simulations of magnetic fusion devices, this approach is motivated by the observation that gradients of the solution along the magnetic field lines are typically much smaller than along a perpendicular direction. In toroidal geometry, field-aligned interpolation consists of a 1D interpolation along the field line, combined with 2D interpolations on the poloidal planes (at the intersections with the field line). A theoretical justification of the method is provided in the simplified context of constant advection on a 2D periodic domain: unconditional stability is proven, and error estimates are given which highlight the advantages of field-aligned interpolation. The same methodology is successfully applied to the solution of the gyrokinetic Vlasov equation, for which we present the ion temperature gradient (ITG) instability as a classical test case: first we solve this in cylindrical geometry (screw-pinch), and next in toroidal geometry (circular Tokamak). In the first case, the algorithm is implemented in Selalib (semi-Lagrangian library), and the numerical simulations provide linear growth rates that are in accordance with the linear dispersion analysis. In the second case, the algorithm is implemented in the Gysela code, and the numerical simulations are benchmarked with those employing the standard (not aligned) scheme. Numerical experiments show that field-aligned interpolation leads to considerable memory savings for the same level of accuracy; substantial savings are also expected in reactor-scale simulations.

We are also currently implementing into SCHNAPS a general transport solver for addressing non-conforming patches in complex geometries. The objective is to be able to design meshes that are able to deal with magnetic aligned geometries. The resulting scheme will be used for solving kinetic equations, of course. But it can also be the building block of a palindromic method applied on curved and non-conforming meshes.