Pseudo-spectral Code Research Articles

Comparison of dealiasing schemes in large-eddy simulation of neutrally stratified atmospheric flows

Abstract. Aliasing errors arise in the multiplication of partial sums, such as those encountered when numerically solving the Navier–Stokes equations, and can be detrimental to the accuracy of a numerical solution. In this work, a performance and cost analysis is proposed for widely used dealiasing schemes in large-eddy simulation, focusing on a neutrally stratified, pressure-driven atmospheric boundary-layer flow. Specifically, the exact 3∕2 rule, the Fourier truncation method, and a high-order Fourier smoothing method are intercompared. Tests are performed within a newly developed mixed pseudo-spectral finite differences large-eddy simulation code, parallelized using a two-dimensional pencil decomposition. A series of simulations are performed at varying resolution, and key flow statistics are intercompared among the considered runs and dealiasing schemes. The three dealiasing methods compare well in terms of first- and second-order statistics for the considered cases, with modest local departures that decrease as the grid stencil is reduced. Computed velocity spectra using the 3∕2 rule and the FS method are in good agreement, whereas the FT method yields a spurious energy redistribution across wavenumbers that compromises both the energy-containing and inertial sublayer trends. The main advantage of the FS and FT methods when compared to the 3∕2 rule is a notable reduction in computational cost, with larger savings as the resolution is increased (15 % for a resolution of 1283, up to a theoretical 30 % for a resolution of 20483).

Thermally driven convection in Li||Bi liquid metal batteries

Liquid Metal Batteries (LMBs) are a promising concept for cheap electrical energy storage at grid level. These are built as a stable density stratification of three liquid layers, with two liquid metals separated by a molten salt. In order to ensure a safe and efficient operation, the understanding of transport phenomena in LMBs is essential. With this motivation we study thermal convection induced by internal heat generation. We consider the electrochemical nature of the cell in order to define the heat balance and the operating parameters. Moreover we develop a simple 1D heat conduction model as well as a fully 3D thermo-fluid dynamics model. The latter is implemented in the CFD library OpenFOAM, extending the volume of fluid solver, and validated against a pseudo-spectral code. Both models are used to study a rectangular 10×10 cm² Li||Bi LMB cell at three different states of charge.

The role of zonal flows in disc gravito-turbulence

The work presented here focuses on the role of zonal flows in the self-sustenance of gravito-turbulence in accretion discs. The numerical analysis is conducted using a bespoke pseudo-spectral code in fully compressible, non-linear conditions. The disc in question, which is modelled using the shearing sheet approximation, is assumed to be self-gravitating, viscous, and thermally diffusive; a constant cooling timescale is also considered. Zonal flows are found to emerge at the onset of gravito-turbulence and they remain closely linked to the turbulent state. A cycle of zonal flow formation and destruction is established, mediated by a slow mode instability (which allows zonal flows to grow) and a non-axisymmetric instability (which disrupts the zonal flow), which is found to repeat numerous times. It is in fact the disruptive action of the non-axisymmetric instability to form new leading and trailing shearing waves, allowing energy to be extracted from the background flow and ensuring the self-sustenance of the gravito-turbulent regime.

The evolution of hyperboloidal data with the dual foliation formalism: mathematical analysis and wave equation tests

A long-standing problem in numerical relativity is the satisfactory treatment of future null-infinity. We propose an approach for the evolution of hyperboloidal initial data in which the outer boundary of the computational domain is placed at infinity. The main idea is to apply the ‘dual foliation’ formalism in combination with hyperboloidal coordinates and the generalized harmonic gauge formulation. The strength of the present approach is that, following the ideas of Zenginoğlu, a hyperboloidal layer can be naturally attached to a central region using standard coordinates of numerical relativity applications. Employing a generalization of the standard hyperboloidal slices, developed by Calabrese et al, we find that all formally singular terms take a trivial limit as we head to null-infinity. A byproduct is a numerical approach for hyperboloidal evolution of nonlinear wave equations violating the null-condition. The height-function method, used often for fixed background spacetimes, is generalized in such a way that the slices can be dynamically ‘waggled’ to maintain the desired outgoing coordinate lightspeed precisely. This is achieved by dynamically solving the eikonal equation. As a first numerical test of the new approach we solve the 3D flat space scalar wave equation. The simulations, performed with the pseudospectral bamps code, show that outgoing waves are cleanly absorbed at null-infinity and that errors converge away rapidly as resolution is increased.

Evolutions of centered Brill waves with a pseudospectral method

The pseudospectral code bamps is used to evolve axisymmetric gravitational waves. We consider a one-parameter family of Brill wave initial data, taking the seed function and strength parameter of Holz et. al. A careful comparison is made to earlier work. Our results are mostly in agreement with the literature, but we do find that some amplitudes reported elsewhere as subcritical evolve to form apparent horizons. Related to this point we find that by altering the slicing condition, the position of the peak of the Kretschmann scalar in these supercritical data can be controlled so that it appears away from the symmetry axis before the method fails, demonstrating that such behavior is at least partially a coordinate effect. We are able to tune the strength parameter to an interval of range $1-A_\star/A\simeq10^{-6}$ around the onset of apparent horizon formation. In this regime we find that large spikes appear in the Kretschmann scalar on the symmetry axis but away from the origin. From the supercritical side disjoint apparent horizons form around these spikes. On the subcritical side, down to this range, evidence of power-law scaling of the Kretschmann scalar is not conclusive, but the data can be fitted as a power-law with periodic wiggle.

Scaling of a Fast Fourier Transform and a pseudo-spectral fluid solver up to 196608 cores

In this paper we present scaling results of a FFT library, FFTK, and a pseudospectral code, Tarang, on grid resolutions up to 81923 grid using 65536 cores of Blue Gene/P and 196608 cores of Cray XC40 supercomputers. We observe that communication dominates computation, more so on the Cray XC40. The computation time scales as Tcomp∼p−1, and the communication time as Tcomm∼n−γ2 with γ2 ranging from 0.7 to 0.9 for Blue Gene/P, and from 0.43 to 0.73 for Cray XC40. FFTK, and the fluid and convection solvers of Tarang exhibit weak as well as strong scaling nearly up to 196608 cores of Cray XC40. We perform a comparative study of the performance on the Blue Gene/P and Cray XC40 clusters.

Influence of the temperature-dependent viscosity on convective flow in the radial force field.

The numerical investigation of convective flows in the radial force field caused by an oscillating electric field between spherical surfaces has been performed. A temperature difference (T_{1}>T_{2}) as well as a radial force field triggers a fluid flow similar to the Rayleigh-Bénard convection. The onset of convective flow has been studied by means of the linear stability analysis as a function of the radius ratio η=R_{1}/R_{2}. The influence of the temperature-dependent viscosity has been investigated in detail. We found that a varying viscosity contrast β=ν(T_{2})/ν(T_{1}) between β=1 (constant viscosity) and β=50 decreases the critical Rayleigh number by a factor of 6. Additionally, we perform a bifurcation analysis based on numerical simulations which have been calculated using a modified pseudospectral code. Numerical results have been compared with the GeoFlow experiment which is located on the International Space Station (ISS). Nonturbulent three-dimensional structures are found in the numerically predicted parameter regime. Furthermore, we observed multiple stable solutions in both experiments and numerical simulations, respectively.

A New View on the Maximum Mass of Differentially Rotating Neutron Stars

Abstract We study the main astrophysical properties of differentially rotating neutron stars described as stationary and axisymmetric configurations of a moderately stiff Γ = 2 polytropic fluid. The high level of accuracy and of stability of our relativistic multidomain pseudo-spectral code enables us to explore the whole solution space for broad ranges of the degree of differential rotation, but also of the stellar density and oblateness. Staying within an astrophysically motivated range of rotation profiles, we investigate the characteristics of neutron stars with maximal mass for all types of families of differentially rotating relativistic objects identified in a previous article. We find that the maximum mass depends on both the degree of differential rotation and the type of solution. It turns out that the maximum allowed mass can be up to 4 times higher than what it is for nonrotating stars with the same equation of state. Such values are obtained for a modest degree of differential rotation but for one of the newly discovered types of solutions. Since such configurations of stars are not that extreme, this result may have important consequences for the gravitational wave signal expected from coalescing neutron star binaries or from some supernova events.

TURBULENT CHEMICAL DIFFUSION IN CONVECTIVELY BOUNDED CARBON FLAMES

ABSTRACT It has been proposed that mixing induced by convective overshoot can disrupt the inward propagation of carbon deflagrations in super-asymptotic giant branch stars. To test this theory, we study an idealized model of convectively bounded carbon flames with 3D hydrodynamic simulations of the Boussinesq equations using the pseudo-spectral code Dedalus. Because the flame propagation timescale is much longer than the convection timescale, we approximate the flame as fixed in space, and only consider its effects on the buoyancy of the fluid. By evolving a passive scalar field, we derive a turbulent chemical diffusivity produced by the convection as a function of height, . Convection can stall a flame if the chemical mixing timescale, set by the turbulent chemical diffusivity, , is shorter than the flame propagation timescale, set by the thermal diffusivity, κ, i.e., when . However, we find for most of the flame because convective plumes are not dense enough to penetrate into the flame. Extrapolating to realistic stellar conditions, this implies that convective mixing cannot stall a carbon flame and that “hybrid carbon–oxygen–neon” white dwarfs are not a typical product of stellar evolution.

Initial-data contribution to the error budget of gravitational waves from neutron-star binaries

As numerical calculations of inspiralling neutron-star binaries reach values of accuracy that are comparable with those of binary black holes, a fine budgeting of the various sources of error becomes increasingly important. Among such sources, the initial data is normally not accounted for, the rationale being that the error on the initial spacelike hypersurface is always far smaller than the one gained during the evolution. We here consider critically this assumption and perform a comparative analysis of the gravitational waveforms relative to essentially the same physical binary configuration when computed with two different initial-data codes, and then evolved with the same evolution code. More specifically, we consider the evolution of irrotational neutron-star binaries computed either with the pseudo-spectral code \lorene{}, or with the newly developed finite-difference code \cocal{}; both sets of initial data are subsequently evolved with the high-order evolution code \whiskythc{}. In this way we find that despite the initial data shows global (local) differences that are $\lesssim 0.02\%\ (1\%)$, the gravitational-wave phase at the merger time differs by $\sim 0.5$ radians after $\sim 3$ orbits, a surprisingly large value. Our results highlight the highly nonlinear impact that errors in the initial data can have on the subsequent evolution and the importance of using exactly the same initial data when comparative studies are done.

Variational integrators for reduced magnetohydrodynamics

Reduced magnetohydrodynamics is a simplified set of magnetohydrodynamics equations with applications to both fusion and astrophysical plasmas, possessing a noncanonical Hamiltonian structure and consequently a number of conserved functionals. We propose a new discretisation strategy for these equations based on a discrete variational principle applied to a formal Lagrangian. The resulting integrator preserves important quantities like the total energy, magnetic helicity and cross helicity exactly (up to machine precision). As the integrator is free of numerical resistivity, spurious reconnection along current sheets is absent in the ideal case. If effects of electron inertia are added, reconnection of magnetic field lines is allowed, although the resulting model still possesses a noncanonical Hamiltonian structure. After reviewing the conservation laws of the model equations, the adopted variational principle with the related conservation laws is described both at the continuous and discrete level. We verify the favourable properties of the variational integrator in particular with respect to the preservation of the invariants of the models under consideration and compare with results from the literature and those of a pseudo-spectral code.

Formation and evolution of vortices in a collisional strongly coupled dusty plasma

Formation and evolution of vortices are studied in a collisional strongly coupled dusty plasma in the framework of a Generalized Hydrodynamic model (GH). Here we mainly present the nonlinear dynamical response of this strongly coupled system in presence of dust-neutral collisional drag. It is shown that the interplay between the nonlinear elastic stress and the dust-neutral collisional drag results in the generation of non-propagating monopole vortex for some duration before it starts to propagate like transverse shear wave. It is also found that the interaction between two unshielded monopole vortices having both same (co-rotating) and opposite (counter rotating) rotations result in the formation of two propagating dipole vortices of equal and unequal strength respectively. These results will provide some new understanding on the transport properties in such a strongly coupled system. The numerical simulation is carried out using a de-aliased doubly periodic pseudo-spectral code with Runge–Kutta–Gill time integrator.

Pseudospectral method for gravitational wave collapse

We present a new pseudospectral code, bamps, for numerical relativity written with the evolution of collapsing gravitational waves in mind. We employ the first order generalized harmonic gauge formulation. The relevant theory is reviewed and the numerical method is critically examined and specialized for the task at hand. In particular we investigate formulation parameters, gauge and constraint preserving boundary conditions well-suited to non-vanishing gauge source functions. Different types of axisymmetric twist-free moment of time symmetry gravitational wave initial data are discussed. A treatment of the axisymmetric apparent horizon condition is presented with careful attention to regularity on axis. Our apparent horizon finder is then evaluated in a number of test cases. Moving on to evolutions, we investigate modifications to the generalized harmonic gauge constraint damping scheme to improve conservation in the strong field regime. We demonstrate strong-scaling of our pseudospectral penalty code. We employ the Cartoon method to efficiently evolve axisymmetric data in our 3+1 dimensional code. We perform test evolutions of Schwarzschild perturbed by gravitational waves and by gauge pulses, both to demonstrate the use of our blackhole excision scheme and for comparison with earlier results. Finally numerical evolutions of supercritical Brill waves are presented to demonstrate durability of the excision scheme for the dynamical formation of a blackhole.

High performance Python for direct numerical simulations of turbulent flows

Direct Numerical Simulations (DNS) of the Navier Stokes equations is an invaluable research tool in fluid dynamics. Still, there are few publicly available research codes and, due to the heavy number crunching implied, available codes are usually written in low-level languages such as C/C++ or Fortran. In this paper we describe a pure scientific Python pseudo-spectral DNS code that nearly matches the performance of C++ for thousands of processors and billions of unknowns. We also describe a version optimized through Cython, that is found to match the speed of C++. The solvers are written from scratch in Python, both the mesh, the MPI domain decomposition, and the temporal integrators. The solvers have been verified and benchmarked on the Shaheen supercomputer at the KAUST supercomputing laboratory, and we are able to show very good scaling up to several thousand cores.A very important part of the implementation is the mesh decomposition (we implement both slab and pencil decompositions) and 3D parallel Fast Fourier Transforms (FFT). The mesh decomposition and FFT routines have been implemented in Python using serial FFT routines (either NumPy, pyFFTW or any other serial FFT module), NumPy array manipulations and with MPI communications handled by MPI for Python (mpi4py). We show how we are able to execute a 3D parallel FFT in Python for a slab mesh decomposition using 4 lines of compact Python code, for which the parallel performance on Shaheen is found to be slightly better than similar routines provided through the FFTW library. For a pencil mesh decomposition 7 lines of code is required to execute a transform.

Influence of the Hall effect and electron inertia in collisionless magnetic reconnection

We study the role of the Hall current and electron inertia in collisionless magnetic reconnection within the framework of full two-fluid MHD. At spatial scales smaller than the electron inertial length, a topological change of magnetic field lines exclusively due to the electron inertia becomes possible. Assuming stationary conditions, we derive a theoretical scaling for the reconnection rate, which is simply proportional to the Hall parameter. Using a pseudo-spectral code with no dissipative effects, our numerical results confirm this theoretical scaling. In particular, for a sequence of different Hall parameter values, our numerical results show that the width of the current sheet is independent of the Hall parameter, while its thickness is of the order of the electron inertial range, thus confirming that the stationary reconnection rate is proportional to the Hall parameter.

A validated non-linear Kelvin–Helmholtz benchmark for numerical hydrodynamics

The nonlinear evolution of the Kelvin-Helmholtz instability is a popular test for code verification. To date, most Kelvin-Helmholtz problems discussed in the literature are ill-posed: they do not converge to any single solution with increasing resolution. This precludes comparisons among different codes and severely limits the utility of the Kelvin-Helmholtz instability as a test problem. The lack of a reference solution has led various authors to assert the accuracy of their simulations based on ad-hoc proxies, e.g., the existence of small-scale structures. This paper proposes well-posed Kelvin-Helmholtz problems with smooth initial conditions and explicit diffusion. We show that in many cases numerical errors/noise can seed spurious small-scale structure in Kelvin-Helmholtz problems. We demonstrate convergence to a reference solution using both Athena, a Godunov code, and Dedalus, a pseudo-spectral code. Problems with constant initial density throughout the domain are relatively straightforward for both codes. However, problems with an initial density jump (which are the norm in astrophysical systems) exhibit rich behavior and are more computationally challenging. In the latter case, Athena simulations are prone to an instability of the inner rolled-up vortex; this instability is seeded by grid-scale errors introduced by the algorithm, and disappears as resolution increases. Both Athena and Dedalus exhibit late-time chaos. Inviscid simulations are riddled with extremely vigorous secondary instabilities which induce more mixing than simulations with explicit diffusion. Our results highlight the importance of running well-posed test problems with demonstrated convergence to a reference solution. To facilitate future comparisons, we include the resolved, converged solutions to the Kelvin-Helmholtz problems in this paper in machine-readable form.

Spectral simulations of an axisymmetric force-free pulsar magnetosphere

A pseudo-spectral method with an absorbing outer boundary is used to solve a set of the time-dependent force-free equations. In the method, both electric and magnetic fields are expanded in terms of the vector spherical harmonic (VSH) functions in spherical geometry and the divergencelessness of magnetic field is analytically enforced by a projection method. Our simulations show that the Deutsch vacuum solution and the Michel monopole solution can be well reproduced by our pseudo-spectral code. Further the method is used to present the time-dependent simulation of the force-free pulsar magnetosphere for an aligned rotator. The simulations show that the current sheet in the equatorial plane can be resolved well, and the obtained spin-down luminosity in the steady state is in good agreement with the value given by Spitkovsky (2006).

Stability of exact force-free electrodynamic solutions and scattering from spacetime curvature

Recently, a family of exact force-free electrodynamic (FFE) solutions was given by Brennan, Gralla and Jacobson, which generalizes earlier solutions by Michel, Menon and Dermer, and other authors. These solutions have been proposed as useful models for describing the outer magnetosphere of conducting stars. As with any exact analytical solution that aspires to describe actual physical systems, it is vitally important that the solution possess the necessary stability. In this paper, we show via fully nonlinear numerical simulations that the aforementioned FFE solutions, despite being highly special in their properties, are nonetheless stable under small perturbations. Through this study, we also introduce a three-dimensional pseudospectral relativistic FFE code that achieves exponential convergence for smooth test cases, as well as two additional well-posed FFE evolution systems in the appendix that have desirable mathematical properties. Furthermore, we provide an explicit analysis that demonstrates how propagation along degenerate principal null directions of the spacetime curvature tensor simplifies scattering, thereby providing an intuitive understanding of why these exact solutions are tractable, i.e. why they are not backscattered by spacetime curvature.

ZOMBIE VORTEX INSTABILITY. I. A PURELY HYDRODYNAMIC INSTABILITY TO RESURRECT THE DEAD ZONES OF PROTOPLANETARY DISKS

There is considerable interest in hydrodynamic instabilities in dead zones of protoplanetary disks as a mechanism for driving angular momentum transport and as a source of particle-trapping vortices to mix chondrules and incubate planetesimal formation. We present simulations with a pseudo-spectral anelastic code and with the compressible code Athena, showing that stably stratified flows in a shearing, rotating box are violently unstable and produce space-filling, sustained turbulence dominated by large vortices with Rossby numbers of order ∼0.2–0.3. This Zombie Vortex Instability (ZVI) is observed in both codes and is triggered by Kolmogorov turbulence with Mach numbers less than ∼0.01. It is a common view that if a given constant density flow is stable, then stable vertical stratification should make the flow even more stable. Yet, we show that sufficient vertical stratification can be unstable to ZVI. ZVI is robust and requires no special tuning of boundary conditions, or initial radial entropy or vortensity gradients (though we have studied ZVI only in the limit of infinite cooling time). The resolution of this paradox is that stable stratification allows for a new avenue to instability: baroclinic critical layers. ZVI has not been seen in previous studies of flows in rotating, shearing boxes because those calculations frequently lacked vertical density stratification and/or sufficient numerical resolution. Although we do not expect appreciable angular momentum transport from ZVI in the small domains in this study, we hypothesize that ZVI in larger domains with compressible equations may lead to angular transport via spiral density waves.

Large Eddy Simulation of wind turbine wakes: detailed comparisons of two codes focusing on effects of numerics and subgrid modeling

In this work we report on results from a detailed comparative numerical study from two Large Eddy Simulation (LES) codes using the Actuator Line Model (ALM). The study focuses on prediction of wind turbine wakes and their breakdown when subject to uniform inflow. Previous studies have shown relative insensitivity to subgrid modeling in the context of a finite-volume code. The present study uses the low dissipation pseudo-spectral LES code from Johns Hopkins University (LESGO) and the second-order, finite-volume OpenFOAMcode (SOWFA) from the National Renewable Energy Laboratory. When subject to uniform inflow, the loads on the blades are found to be unaffected by subgrid models or numerics, as expected. The turbulence in the wake and the location of transition to a turbulent state are affected by the subgrid-scale model and the numerics.