Field-Aligned Interpolation for Semi-Lagrangian Gyrokinetic Simulations

Guillaume Latu,Eric Sonnendrücker,Maurizio Ottaviani,Yaman Güçlü,Michel Mehrenberger

doi:10.1007/s10915-017-0509-5

Abstract

This work is devoted to the study of field-aligned interpolation in semi-Lagrangian codes. 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.

Highlights

In a Tokamak, due to the large confining magnetic field, a fast homogenisation of the different physical quantities occurs along the magnetic field lines; this leads to very smooth and small variations along the field lines, whereas the scale length of the variations is very small in a perpendicular direction
A fundamental non-dimensional parameter in magnetic fusion devices is the ratio ρ∗ = ρs/A, where A is the minor radius of the device. (In terms of nondimensional quantities, the minor radius of the device is a = A/ρs and ρ∗ = 1/a.) The number of degrees of freedom needed to represent a poloidal cut of the solution scale with (ρ∗)−2, smaller values of ρ∗ lead to larger numerical simulations
We have described a semi-Lagrangian method based on field-aligned interpolation, for the solution of the gyrokinetic Vlasov equation

Summary

Introduction

In a Tokamak, due to the large confining magnetic field, a fast homogenisation of the different physical quantities occurs along the magnetic field lines; this leads to very smooth and small variations along the field lines, whereas the scale length of the variations is very small (comparable to the gyro-radius) in a perpendicular direction. This should be taken into account for more efficient simulations. We are interested here in a thorough numerical investigation of this idea in the context of gyrokinetic simulations using semi-Lagrangian methods. The reader interested in the physics context can consult the review article [4], and exhaustive information about the semi-Lagrangian method which was introduced in the context of gyrokinetic simulations in [5] and the Gysela code are provided in [6]

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Conclusion