Field-Aligned Interpolation for Semi-Lagrangian Gyrokinetic Simulations

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, fieldaligned 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.

Derivative-free high-order uniformly accurate schemes for highly oscillatory systems

“…with x = (x 1 , x 2 , x 3 ) T and α = 0.5. This choice of B is known as the screw-pinch case [23] with |B(x)| = 1. We apply our 3 rd order and 4 th order (AB-MM) methods to (5.5) withN τ = 128.…”

Stability of Semi-Lagrangian schemes of arbitrary odd degree under constant and variable advection speed

“…2.1.1. Proof using a result from [12]. In this subsection, we prove (2.10) via the following technical result from [12, Section 3.2.4].…”

Computing the Double-Gyroaverage Term Incorporating Short-Scale Perturbation and Steep Equilibrium Profile by the Interpolation Algorithm

“…The whole simulations are executed on the ATLAS HPC platform of IRMA by introducing our test modules into SeLaLiB code [32], which is a classic semi-Lagrangian library using cubic splines interpolation [33][34][35][36]. The predictor-corrector method, the Verlet algorithm for computing the characteristics, and the Strang splitting of the advection of Vlasov equation are also implemented [35,[37][38][39][40].…”