Three discontinuous Galerkin schemes for the anisotropic heat conduction equation on non-aligned grids

We present and discuss three discontinuous Galerkin (dG) discretizations for the anisotropic heat conduction equation on non-aligned cylindrical grids. Our most favourable scheme relies on a self-adjoint local dG (LDG) discretization of the elliptic operator. It conserves the energy exactly and converges with arbitrary order. The pollution by numerical perpendicular heat fluxes degrades with superconvergence rates. We compare this scheme with aligned schemes that are based on the flux-coordinate independent approach for the discretization of parallel derivatives. Here, the dG method provides the necessary interpolation. The first aligned discretization can be used in an explicit time-integrator. However, the scheme violates conservation of energy and shows up stagnating convergence rates for very high resolutions. We overcome this partly by using the adjoint of the parallel derivative operator to construct a second self-adjoint aligned scheme. This scheme preserves energy, but reveals unphysical oscillations in the numerical tests, which result in a decreased order of convergence. Both aligned schemes exhibit low numerical heat fluxes into the perpendicular direction. We build our argumentation on various numerical experiments on all three schemes for a general axisymmetric magnetic field, which is closed by a comparison to the aligned finite difference (FD) schemes of References [1,2

The collisional drift wave instability in steep density gradient regimes

The collisional drift wave instability in a straight magnetic field configuration is studied within a full-F gyro-fluid model, which relaxes the Oberbeck-Boussinesq (OB) approximation. Accordingly, we focus our study on steep background density gradients. In this regime we report on corrections by factors of order one to the eigenvalue analysis of former OB approximated approaches as well as on spatially localised eigenfunctions, that contrast strongly with their OB approximated equivalent. Remarkably, non-modal phenomena arise for large density inhomogeneities and for all collisionalities. As a result, we find initial decay and non-modal growth of the free energy and radially localised and sheared growth patterns. The latter non-modal effect sustains even in the nonlinear regime in the form of radially localised turbulence or zonal flow amplitudes.Comment: accepted at Nuclear Fusio

Beyond the Oberbeck-Boussinesq and long wavelength approximation

We present the first simulations of a reduced magnetized plasma model that incorporates both arbitrary wavelength polarization and non-Oberbeck-Boussinesq effects. Significant influence of these two effects on the density, electric potential and ExB vorticity and non-linear dynamics of blobs are reported. Arbitrary wavelength polarization implicates so-called gyro-amplification that compared to a long wavelength approximation leads to highly amplified small-scale ExB vorticity fluctuations. These strongly increase the coherence and lifetime of blobs and alter the motion of the blobs through a faster blob-disintegration. Non-Oberbeck-Boussinesq effects incorporate plasma inertia, which substantially decreases the growth rate and linear acceleration of high amplitude blobs, while the maximum blob velocity is not affected. Finally, we generalize and numerically verify unified scaling laws for blob velocity, acceleration and growth rate that include both ion temperature and arbitrary blob amplitude dependence

Non-Oberbeck-Boussinesq zonal flow generation

Novel mechanisms for zonal flow (ZF) generation for both large relative density fluctuations and background density gradients are presented. In this non-Oberbeck-Boussinesq (NOB) regime ZFs are driven by the Favre stress, the large fluctuation extension of the Reynolds stress, and by background density gradient and radial particle flux dominated terms. Simulations of a nonlinear full-F gyro-fluid model confirm the predicted mechanism for radial ZF propagation and show the significance of the NOB ZF terms for either large relative density fluctuation levels or steep background density gradients

Consistency in Drift-ordered Fluid Equations

We address several concerns related to the derivation of drift-ordered fluid equations. Starting from a fully Galilean invariant fluid system, we show how consistent sets of perturbative drift-fluid equations in the case of a isothermal collisionless fluid can be obtained. Treating all the dynamical fields on equal footing in the singular-drift expansion, we show under what conditions a set of perturbative equations can have a non-trivial quasi-neutral limit. We give a suitable perturbative setup where we provide the full set of perturbative equations for obtaining the first-order corrected fields and show that all the constants of motion are preserved at each order. With the dynamical field variables under perturbative control, we subsequently provide a quantitative analysis by means of numerical simulations. With direct access to first-order corrections the convergence properties are addressed for different regimes of parameter space and the validity of the first-order approximation is discussed in the three settings: cold ions, hot ions and finite charge density.Comment: 22 page

Unified transport scaling laws for plasma blobs and depletions

We study the dynamics of seeded plasma blobs and depletions in an (effective) gravitational field. For incompressible flows the radial center of mass velocity of blobs and depletions is proportional to the square root of their initial cross-field size and amplitude. If the flows are compressible, this scaling holds only for ratios of amplitude to size larger than a critical value. Otherwise, the maximum blob and depletion velocity depends linearly on the initial amplitude and is independent of size. In both cases the acceleration of blobs and depletions depends on their initial amplitude relative to the background plasma density, is proportional to gravity and independent of their cross-field size. Due to their reduced inertia plasma depletions accelerate more quickly than the corresponding blobs. These scaling laws are derived from the invariants of the governing drift-fluid equations and agree excellently with numerical simulations over five orders of magnitude. We suggest an empirical model that unifies and correctly captures the radial acceleration and maximum velocities of both blobs and depletions

Pad\'e-based arbitrary wavelength polarization closures for full-F gyro-kinetic and -fluid models

We propose a solution to the long-standing short wavelength polarization closure shortfall of full-F gyro-fluid models. This is achieved by first finding an appropriate quadratic form of the gyro-fluid moment over the polarization part of the gyro-center Hamiltonian. Secondly, we deduce Pad\'e-based approximations to the latter expression that produce a polarization charge density with the desired order of accuracy and retain linear polarization effects for arbitrary wavelengths. The proposed closures feature proper energy conservation and the anticipated Oberbeck-Boussinesq and long wavelength limits

Reproducibility, accuracy and performance of the Feltor code and library on parallel computer architectures

Feltor is a modular and free scientific software package. It allows developing platform independent code that runs on a variety of parallel computer architectures ranging from laptop CPUs to multi-GPU distributed memory systems. Feltor consists of both a numerical library and a collection of application codes built on top of the library. Its main target are two- and three-dimensional drift- and gyro-fluid simulations with discontinuous Galerkin methods as the main numerical discretization technique. We observe that numerical simulations of a recently developed gyro-fluid model produce non-deterministic results in parallel computations. First, we show how we restore accuracy and bitwise reproducibility algorithmically and programmatically. In particular, we adopt an implementation of the exactly rounded dot product based on long accumulators, which avoids accuracy losses especially in parallel applications. However, reproducibility and accuracy alone fail to indicate correct simulation behaviour. In fact, in the physical model slightly different initial conditions lead to vastly different end states. This behaviour translates to its numerical representation. Pointwise convergence, even in principle, becomes impossible for long simulation times. In a second part, we explore important performance tuning considerations. We identify latency and memory bandwidth as the main performance indicators of our routines. Based on these, we propose a parallel performance model that predicts the execution time of algorithms implemented in Feltor and test our model on a selection of parallel hardware architectures. We are able to predict the execution time with a relative error of less than 25% for problem sizes between 0.1 and 1000 MB. Finally, we find that the product of latency and bandwidth gives a minimum array size per compute node to achieve a scaling efficiency above 50% (both strong and weak)

Reproducibility Strategies for Parallel Preconditioned Conjugate Gradient

The Preconditioned Conjugate Gradient method is often used in numerical simulations. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we aim at a twofold goal: enhance the accuracy of the solver but also ensure its reproducibility in a message-passing implementation. We design and employ various strategies starting from the ExBLAS approach (through preserving every bit of information until final rounding) to its more lightweight performance-oriented variant (through expanding the intermediate precision). These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these strategies on modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of only 29 % (ExBLAS) and 4 % (lightweight) on 768 processes