Towards Semi-Implicit Methods for the GX Gyrokinetics Code
Patrick Kim, Princeton University
The total power output of nuclear fusion devices such as tokamaks and stellarators is often limited by the anomalous transport of heat and particles by plasma turbulence, which is well described in such devices by the gyrokinetic model [1-3]. Unfortunately, transport simulations are often limited by the large computational cost of gyrokinetic codes. Recently, the GX gyrokinetics code [4-5] has demonstrated dramatic improvements in speed due to its flexible pseudo-spectral representation in velocity space and native GPU implementation. However, GX uses a purely explicit time-integration scheme, and so suffers from a greatly reduced time-step when kinetically evolving electrons as they have a thermal speed 60 times greater than that of deuterium ions. In this work, we begin implementing semi-implicit methods to address the stiff electron terms in the gyrokinetic equation using third-order additive Runge-Kutta methods from [6]. In particular, we focus on terms associated with the streaming of electrons along a uniform magnetic field. These terms result in a tri-diagonal system that can easily be inverted at each time-step. Overall it allows us to increase the maximum stable time-step by up to a factor of 60 with minimal increase in computational cost per time-step. Finally, we present preliminary work on inverting stiff terms that arise when introducing an inhomogeneous magnetic field typical of tokamaks and stellarators.
[1] Catto, "Linearized gyro-kinetics", Plasma Physics, 1978.
[2] Antonsen and Lane, "Kinetic equations for low frequency instabilities in inhomogeneous plasmas", The Physics of Fluids, 1980.
[3] Frieman and Chen, "Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria", The Physics of Fluids, 1982.
[4] Mandell et al., "Laguerre-Hermite pseudo-spectral velocity formulation of gyrokinetics", J. Plasma Physics, 2018.
[5] Mandell et al., "GX: a GPU-native gyrokinetic turbulence code for tokamak and stellarator design", 2024
[6] Conde et al., "Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order", J. Scientific Computing, 2017.