Neutrino Transport Research Articles

General relativistic neutrino transport with spectral methods

Supernovae numerical simulations are highly demanding in computational power, yet they are also rapidly increasing in complexity. There is now a very high demand in new methods for neutrino treatment, due to the fact that neutrino is at the same time a crucial ingredient and the most expensive one in computational resources. A possibility to explore is the treatment by spectral methods, because they are known to require less resources. It is the purpose of a new code, named Lorene's Ghost (for Lorene's gravitational handling of spectral transport), developed to treat the problem of neutrino transport in supernovae. The code can handle full phase-space dependent problems (6D + time), retains the full energy dependence of the neutrinos, and can also handle general relativity in the conformally flat approximation. Several points of the formalism needed to obtain a formulation of the general relativistic Boltzmann equation adapted to our case are discussed, as well as some technical aspects of spectral methods.

Non-radial instabilities and progenitor asphericities in core-collapse supernovae

Since core-collapse supernova simulations still struggle to produce robust neutrino-driven explosions in 3D, it has been proposed that asphericities caused by convection in the progenitor might facilitate shock revival by boosting the activity of non-radial hydrodynamic instabilities in the post-shock region. We investigate this scenario in depth using 42 relativistic 2D simulations with multi-group neutrino transport to examine the effects of velocity and density perturbations in the progenitor for different perturbation geometries that obey fundamental physical constraints (like the anelastic condition). As a framework for analysing our results, we introduce semi-empirical scaling laws relating neutrino heating, average turbulent velocities in the gain region, and the shock deformation in the saturation limit of non-radial instabilities. The squared turbulent Mach number, <Ma^2>, reflects the violence of aspherical motions in the gain layer, and explosive runaway occurs for <Ma^2>~0.3, corresponding to a reduction of the critical neutrino luminosity by ~25% compared to 1D. In the light of this theory, progenitor asphericities aid shock revival mainly by creating anisotropic mass flux onto the shock: Differential infall efficiently converts velocity perturbations in the progenitor into density perturbations (Delta rho/rho) at the shock of the order of the initial convective Mach number Ma. The anisotropic mass flux and ram pressure deform the shock and thereby amplify post-shock turbulence. Large-scale (l=2,l=1) modes prove most conducive to shock revival, whereas small-scale perturbations require unrealistically high convective Mach numbers. Initial density perturbations in the progenitor are only of order Ma^2 and therefore play a subdominant role.

Comprehensive nucleosynthesis analysis for ejecta of compact binary mergers

We present the first comprehensive study of r-process element nucleosynthesis in the ejecta of compact binary mergers (CBMs) and their relic black-hole (BH)-torus systems. The evolution of the BH-accretion tori is simulated for seconds with a Newtonian hydrodynamics code including viscosity effects, pseudo-Newtonian gravity for rotating BHs, and an energy-dependent two-moment closure scheme for the transport of electron neutrinos and antineutrinos. The investigated cases are guided by relativistic double neutron star (NS-NS) and NS-BH merger models, producing ~3-6 Msun BHs with rotation parameters of A~0.8 and tori of 0.03-0.3 Msun. Our nucleosynthesis analysis includes the dynamical (prompt) ejecta expelled during the CBM phase and the neutrino and viscously driven outflows of the relic BH-torus systems. While typically ~20-25% of the initial accretion-torus mass are lost by viscously driven outflows, neutrino-powered winds contribute at most another ~1%, but neutrino heating enhances the viscous ejecta significantly. Since BH-torus ejecta possess a wide distribution of electron fractions (0.1-0.6) and entropies, they produce heavy elements from A~80 up to the actinides, with relative contributions of A>130 nuclei being subdominant and sensitively dependent on BH and torus masses and the exact treatment of shear viscosity. The combined ejecta of CBM and BH-torus phases can reproduce the solar abundances amazingly well for A>90. Varying contributions of the torus ejecta might account for observed variations of lighter elements with 40<Z<56 relative to heavier ones, and a considerable reduction of the prompt ejecta compared to the torus ejecta, e.g. in highly asymmetric NS-BH mergers, might explain the composition of heavy-element deficient stars.

TWO-DIMENSIONAL CORE-COLLAPSE SUPERNOVA MODELS WITH MULTI-DIMENSIONAL TRANSPORT

We present new two-dimensional (2D) axisymmetric neutrino radiation/hydrodynamic models of core-collapse supernova (CCSN) cores. We use the CASTRO code, which incorporates truly multi-dimensional, multi-group, flux-limited diffusion (MGFLD) neutrino transport, including all relevant terms. Our main motivation for carrying out this study is to compare with recent 2D models produced by other groups who have obtained explosions for some progenitor stars and with recent 2D VULCAN results that did not incorporate terms. We follow the evolution of 12, 15, 20, and 25 solar-mass progenitors to approximately 600 ms after bounce and do not obtain an explosion in any of these models. Though the reason for the qualitative disagreement among the groups engaged in CCSN modeling remains unclear, we speculate that the simplifying "ray-by-ray" approach employed by all other groups may be compromising their results. We show that "ray-by-ray" calculations greatly exaggerate the angular and temporal variations of the neutrino fluxes, which we argue are better captured by our multi-dimensional MGFLD approach. On the other hand, our 2D models also make approximations, making it difficult to draw definitive conclusions concerning the root of the differences between groups. We discuss some of the diagnostics often employed in the analyses of CCSN simulations and highlight the intimate relationship between the various explosion conditions that have been proposed. Finally, we explore the ingredients that may be missing in current calculations that may be important in reproducing the properties of the average CCSNe, should the delayed neutrino-heating mechanism be the correct mechanism of explosion.

Fast-time Variations of Supernova Neutrino Fluxes and Detection Perspectives

In the delayed explosion scenario of a core-collapse supernova, the accretion phase shows pronounced convective over-turns and a low-multipole hydrodynamic instability, the so-called standing accretion shock instability (SASI). Neutrino signal variations from the first full-scale three-dimensional core-collapse supernova simulations with sophisticated neutrino transport are presented as well as their detection perspectives in IceCube and Hyper-Kamiokande.

The influence of inelastic neutrino interactions with light clusters on core-collapse supernova simulations

We perform numerical experiments to investigate the influence of inelastic neutrino reactions with light clusters in hot nuclear matter on core-collapse supernova simulations. These interactions have been neglected in most hydrodynamical supernova simulations. The neutrino absorptions and inelastic interactions with deuterons, tritons, helions and alpha particles are taken into account in the hydrodynamical simulations in addition to the ordinary charged- current interactions with nucleons. Axial symmetry is assumed but no equatorial symmetry is imposed. The time evolutions of shock waves are calculated with a simple light-bulb approximation for the neutrino transport and a multi-nuclei equation of state. We show that the heating rates of deuterons reach as high as ~ 10% of those of nucleons around the bottom of the gain region. On the other hand, alpha particles heat the matter near the shock wave, which is important when the shock wave expands and density and temperature of matter become low. It is also found that the models with heating by light clusters have different evolutions from those without it in non-linear evolution phase. The matter in the gain region has various densities and temperatures and there appear regions that are locally rich in deuterons and alpha particles. These results indicate that the inelastic reactions of light clusters, especially deuterons, should be incorporated in the simulations of core-collapse supernovae.

Magnetic field amplification and magnetically supported explosions of collapsing, non-rotating stellar cores

We study the amplification of magnetic fields in the collapse and the post-bounce evolution of the core of a non-rotating star of 15 solar masses in axisymmetry. To this end, we solve the coupled equations of magnetohydrodynamics and neutrino transport in the two-moment approximation. The pre-collapse magnetic field is strongly amplified by compression in the infall. Initial fields of the order of 1010 G translate into proto-neutron star fields similar to the ones observed in pulsars, while stronger initial fields yield magnetar-like final field strengths. After core bounce, the field is advected through the hydrodynamically unstable neutrino-heating layer, where non-radial flows due to convection and the standing accretion shock instability amplify the field further. Consequently, the resulting amplification factor of order five is the result of the number of small-eddy turnovers taking place within the time scale of advection through the post-shock layer. Due to this limit, most of our models do not reach equipartition between kinetic and magnetic energy and, consequently, evolve similarly to the non-magnetic case, exploding after about 800 ms when a single or few high-entropy bubbles persist over several dynamical time scales. In the model with the strongest initial field we studied, \$10^{12}\$ G, for which equipartition between flow and field is achieved, the magnetic tension favours a much earlier development of such long-lived high-entropy bubbles and enforces a fairly ordered large-scale flow pattern. Consequently, this model, after exhibiting very regular shock oscillations, explodes much earlier than non-magnetic ones.

Neutrino emission characteristics and detection opportunities based on three-dimensional supernova simulations

The neutrino emission characteristics of the first full-scale three-dimensional supernova simulations with sophisticated three-flavor neutrino transport for three models with masses 11.2, 20 and 27 M_sun are evaluated in detail. All the studied progenitors show the expected hydrodynamical instabilities in the form of large-scale convective overturn. In addition, the recently identified LESA phenomenon (lepton-number emission self-sustained asymmetry) is generic for all our cases. Pronounced SASI (standing accretion-shock instability) activity appears in the 20 and 27 M_sun cases, partly in the form of a spiral mode, inducing large but direction and flavor-dependent modulations of neutrino emission. These modulations can be clearly identified in the existing IceCube and future Hyper-Kamiokande detectors, depending on distance and detector location relative to the main SASI sloshing direction.

SELF-SUSTAINED ASYMMETRY OF LEPTON-NUMBER EMISSION: A NEW PHENOMENON DURING THE SUPERNOVA SHOCK-ACCRETION PHASE IN THREE DIMENSIONS

During the stalled-shock phase of our 3D hydrodynamical core-collapse simulations with energy-dependent, 3-flavor neutrino transport, the lepton-number flux (nue minus antinue) emerges predominantly in one hemisphere. This novel, spherical-symmetry breaking neutrino-hydrodynamical instability is termed LESA for "Lepton-number Emission Self-sustained Asymmetry." While the individual nue and antinue fluxes show a pronounced dipole pattern, the heavy-flavor neutrino fluxes and the overall luminosity are almost spherically symmetric. Initially, LESA seems to develop stochastically from convective fluctuations, it exists for hundreds of milliseconds or more, and it persists during violent shock sloshing associated with the standing accretion shock instability. The nue minus antinue flux asymmetry originates mainly below the neutrinosphere in a region of pronounced proto-neutron star (PNS) convection, which is stronger in the hemisphere of enhanced lepton-number flux. On this side of the PNS, the mass-accretion rate of lepton-rich matter is larger, amplifying the lepton-emission asymmetry, because the spherical stellar infall deflects on a dipolar deformation of the stalled shock. The increased shock radius in the hemisphere of less mass accretion and minimal lepton-number flux (antinue flux maximum) is sustained by stronger convection on this side, which is boosted by stronger neutrino heating because the average antinue energy is higher than the average nue energy. Asymmetric heating thus supports the global deformation despite extremely nonstationary convective overturn behind the shock. While these different elements of LESA form a consistent picture, a full understanding remains elusive at present. There may be important implications for neutrino-flavor oscillations, the neutron-to-proton ratio in the neutrino-heated supernova ejecta, and neutron-star kicks, which remain to be explored.

MODA: a new algorithm to compute optical depths in multidimensional hydrodynamic simulations

We introduce a new algorithm for the calculation of multidimensional optical depths in approximate radiative transport schemes, equally applicable to neutrinos and photons. Motivated by (but not limited to) neutrino transport in three-dimensional simulations of core-collapse supernovae and neutron star mergers, our method makes no assumptions about the geometry of the matter distribution, apart from expecting optically transparent boundaries. Based on local information about opacities, the algorithm figures out an escape route that tends to minimize the optical depth without assuming any pre-defined paths for radiation. Its adaptivity makes it suitable for a variety of astrophysical settings with complicated geometry (e.g., core-collapse supernovae, compact binary mergers, tidal disruptions, star formation, etc.). We implement the MODA algorithm into both a Eulerian hydrodynamics code with a fixed, uniform grid and into an SPH code where we make use a tree structure that is otherwise used for searching neighbours and calculating gravity. In a series of numerical experiments, we compare the MODA results with analytically known solutions. We also use snapshots from actual 3D simulations and compare the results of MODA with those obtained with other methods such as the global and local ray-by-ray method. It turns out that MODA achieves excellent accuracy at a moderate computational cost. In an appendix we also discuss implementation details and parallelization strategies.

Supernova explosion mechanism taking into account large-scale convection and neutrino transport

Two types of supernovae are considered: thermonuclear supernovae, whose explosions are due to thermonuclear energy, and core-collapse supernovae, whose explosions are due to the gravitational energy of collapsing stars released in the form of neutrinos. Numerical models of supernovae are discussed. Themain problem in devising supernova explosion mechanisms is producing the energy required to disperse the envelope. In theoretical models, it is necessary to solve multi-dimensional problems involving complex physics (3D gas dynamics, neutrino transport, large-scale convective instability, and other important physical processes). In recent years, the development of large-scale convection during supernova explosions has been reconsidered. Self-consistent problems problems in three-dimensional, gas-dynamical instability have been considered. Two-dimensional gas-dynamical calculations taking into account neutrino absorption in the envelope have been performed. The spherically symmetric collapse and neutrino transport were calculated including all reactions, leading to a new understanding of possible paths for the development of supernova theory. The main emphasis is placed on the neutrino transport and the basis for promising multidimensional models taking into account large-scale convective instability.

PRODUCTION OF ALL THE r -PROCESS NUCLIDES IN THE DYNAMICAL EJECTA OF NEUTRON STAR MERGERS

Recent studies suggest that binary neutron star (NS-NS) mergers robustly produce the heavy r-process nuclei above the atomic mass number A ~ 130 because of their ejecta consisting of almost pure neutrons (electron fraction of Y_e < 0.1). However, little production of the lighter r-process nuclei (A = 90-120) conflicts with the spectroscopic results of r-process-enhanced Galactic halo stars. We present, for the first time, the result of nucleosynthesis calculations based on the fully general-relativistic simulation of a NS-NS merger with approximate neutrino transport. It is found that the bulk of the dynamical ejecta are appreciably shock-heated and neutrino-processed, resulting in a wide range of Y_e (= 0.09-0.45). The mass-averaged abundance distribution of calculated nucleosynthesis yields is in reasonable agreement with the full-mass range (A = 90-240) of the solar r-process curve. This implies, if our model is representative of such events, that the dynamical ejecta of NS-NS mergers can be the origin of the Galactic r-process nuclei. Our result also shows that the radioactive heating after ~ 1 day from the merging, giving rise to r-process-powered transient emission, is dominated by the beta-decays of several species close to stability with precisely measured half-lives. This implies that the total radioactive heating rate for such an event can be well constrained within about a factor of two if the ejected material has a solar-like r-process pattern.

A NEW MULTI-DIMENSIONAL GENERAL RELATIVISTIC NEUTRINO HYDRODYNAMICS CODE FOR CORE-COLLAPSE SUPERNOVAE. IV. THE NEUTRINO SIGNAL

Considering general relativistic, two-dimensional (2D) supernova (SN) explosion models of progenitor stars between 8.1 and 27 solar masses, we systematically analyze the properties of the neutrino emission from core collapse and bounce to the post-explosion phase. The models were computed with the Vertex-CoCoNuT code, using three-flavor, energy-dependent neutrino transport in the ray-by-ray-plus approximation. Our results confirm the close similarity of the mean energies of electron antineutrinos and heavy-lepton neutrinos and even their crossing during the accretion phase for stars with M>10 M_sun as observed in previous 1D and 2D simulations with state-of-the-art neutrino transport. We establish a roughly linear scaling of the electron antineutrino mean energy with the proto-neutron star (PNS) mass, which holds in time as well as for different progenitors. Convection inside the PNS affects the neutrino emission on the 10-20% level, and accretion continuing beyond the onset of the explosion prevents the abrupt drop of the neutrino luminosities seen in artificially exploded 1D models. We demonstrate that a wavelet-based time-frequency analysis of SN neutrino signals in IceCube will offer sensitive diagnostics for the SN core dynamics up to at least ~10kpc distance. Strong, narrow-band signal modulations indicate quasi-periodic shock sloshing motions due to the standing accretion shock instability (SASI), and the frequency evolution of such "SASI neutrino chirps" reveals shock expansion or contraction. The onset of the explosion is accompanied by a shift of the modulation frequency below 40-50Hz, and post-explosion, episodic accretion downflows will be signaled by activity intervals stretching over an extended frequency range in the wavelet spectrogram.

A COMPARISON OF TWO- AND THREE-DIMENSIONAL NEUTRINO-HYDRODYNAMICS SIMULATIONS OF CORE-COLLAPSE SUPERNOVAE

We present numerical results on two- (2D) and three-dimensional (3D) hydrodynamic core-collapse simulations of an 11.2$M_\odot$ star. By changing numerical resolutions and seed perturbations systematically, we study how the postbounce dynamics is different in 2D and 3D. The calculations were performed with an energy-dependent treatment of the neutrino transport based on the isotropic diffusion source approximation scheme, which we have updated to achieve a very high computational efficiency. All the computed models in this work including nine 3D models and fifteen 2D models exhibit the revival of the stalled bounce shock, leading to the possibility of explosion. All of them are driven by the neutrino-heating mechanism, which is fostered by neutrino-driven convection and the standing-accretion-shock instability (SASI). Reflecting the stochastic nature of multi-dimensional (multi-D) neutrino-driven explosions, the blast morphology changes from models to models. However, we find that the final fate of the multi-D models whether an explosion is obtained or not, is little affected by the explosion stochasticity. In agreement with some previous studies, higher numerical resolutions lead to slower onset of the shock revival in both 3D and 2D. Based on the self-consistent supernova models leading to the possibility of explosions, our results systematically show that the revived shock expands more energetically in 2D than in 3D.

Essential ingredients in core-collapse supernovae

Carrying 1044 joules of kinetic energy and a rich mix of newly synthesized atomic nuclei, core-collapse supernovae are the preeminent foundries of the nuclear species which make up our solar system and ourselves. Signaling the inevitable death of a massive star, and the birth of a neutron star or black hole, core-collapse supernovae combine physics over a wide range in spatial scales, from kilometer-sized hydrodynamic motions (eventually growing to gigameter scale) down to femtometer-scale nuclear reactions. We will discuss our emerging understanding of the convectively-unstable, neutrino-driven explosion mechanism, based on increasingly realistic neutrino radiation hydrodynamic simulations that include progressively better nuclear and particle physics. Multi-dimensional models with spectral neutrino transport from several research groups, which slowly develop successful explosions for a range of progenitors, have recently motivated changes in our understanding of the neutrino reheating mechanism. In a similar fashion, improvements in nuclear physics, most notably explorations of weak interactions on nuclei and the nuclear equation of state, continue to refine our understanding of the births of neutron stars and the supernovae that result. Recent progress on both the macroscopic and microscopic effects that affect core-collapse supernovae are discussed.

Neutrino scattering from hydrodynamic modes in hot and dense neutron matter

We calculate the scattering rate of low-energy neutrinos in hot and dense neutron matter encountered in neutrons stars and supernovae in the hydrodynamic regime. We find that the Brillouin peak, associated with the sound mode, and the Rayleigh peak, associated with the thermal diffusion mode, dominate the dynamic structure factor. Although the total scattering cross section is constrained by the compressibility sum rule, the differential cross section calculated using the hydrodynamic response function differs from results obtained in approximate treatments often used in astrophysics such as random phase approximations. We identified these differences and discuss their implications for neutrino transport in supernovae.

Neutrinos in core-collapse supernovae and nucleosynthesis

Neutrino interactions are a key component of the mechanism for supernova explosions. They also play an important role in the associated nucleosynthesis. Nevertheless, a number of issues remain to be resolved. In this review we summarize research by our group and others on some of these issues. Foremost is the role of neutrinos in the explosion itself. Much effort is now invested in understanding the transport of neutrinos from the core and the role of neutrino heated convection in the outer envelopes. In addition to the neutrino transport issues, we review some current topics in neutrino-induced nucleosynthesis (the ν- and νp-processes). We review some current issues regarding the synthesis of the ν-process isotopes 180Ta and 138La and summarize the application of the ν-process chronometers, 136, 138Ce, 138La, and 92Nb. We also consider the light ν-process isotopes 7Li and 11B and summarize how neutrino oscillations and nucleosynthesis might constrain the neutrino mass hierarchy. We will also review the current dilemma regarding the nucleosynthesis of heavy elements by rapid neutron capture in the neutrino energized wind above the nascent neutron star. We also review the crucial role of neutrino reactions and heating above the high-temperature accretion disc around the black hole of a failed supernova (collapsar model). This collapsar model for long-duration gamma ray bursts is a possible site for r-process nucleosynthesis. We present numerical r-process calculations in the context of a magnetohydrodynamics + neutrino pair heated collapsar simulation. Neutrino heating of the jet is crucial for achieving the required relativistic outflow and at the same time generating material with a high neutron excess. During the late time evolution of the jet an r-process-like abundance distribution is formed within neutrino heated regions of the relativistic outflowing jet. Indeed, sufficient mass is ejected within the flow to account for the observed r-process abundance distribution along with the large dispersion in r-process elements observed in metal-poor halo stars. Finally, we discuss the nuclear physics issues and possible role of relic supernova neutrinos in resolving the supernova rate problem.

Supernova neutrinos and nucleosynthesis

Observations of metal-poor stars indicate that at least two different nucleosynthesis sites contribute to the production of r-process elements. One site is responsible for the production of light r-process elements Z ≲ 50, while the other produces the heavy r-process elements. We have analyzed recent observations of metal-poor stars selecting only stars that are enriched in light r-process elements and poor in heavy r-process elements. We find a strong correlation between the observed abundances of the N = 50 elements (Sr, Y and Zr) and Fe. It suggest that neutrino-driven winds from core-collapse supernova are the main site for the production of these elements. We explore this possibility by performing nucleosynthesis calculations based on long-term Boltzmann neutrino transport simulations. We use an equation of state that reproduces recent constrains on the nuclear symmetry energy. We predict that the early ejecta is neutron rich with Ye ∼ 0.48, it becomes proton rich around 4 s and reaches Ye = 0.586 at 9 s when our simulation stops. The nucleosynthesis in this model produces elements between Zn and Mo, including 92Mo. The elemental abundances are consistent with the observations of the metal-poor star HD 12263. For the elements between Ge and Mo, we produce mainly the neutron-deficient isotopes. This prediction can be confirmed by observations of isotopic abundances in metal-poor stars. No elements heavier than Mo (Z = 42) and no heavy r-process elements are produced in our calculations.

Gravitational wave signatures from low-mode spiral instabilities in rapidly rotating supernova cores

We study properties of gravitational waves (GWs) from the rotating core collapse of a $15{M}_{\ensuremath{\bigodot}}$ star by performing three-dimensional general-relativistic hydrodynamic simulations with an approximate neutrino transport. By parametrically changing the precollapse angular momentum, we focus on the effects of rotation on the GW signatures in the early postbounce evolution. Regarding three-flavor neutrino transport, we solve the energy-averaged set of radiation energy and momentum based on the Thorne's momentum formalism. In addition to the gravitational quadrupole radiation from matter motions, we take into account GWs from anisotropic neutrino emission. With these computations, our results present supporting evidence for the previous anticipation that nonaxisymmetric instabilities play an essential role in determining the postbounce GW signatures. During prompt convection, we find that the waveforms show narrow-band and highly quasiperiodic signals which persist until the end of simulations. We point out that such features reflect the growth of the one-armed spiral modes. The typical frequency of the quasiperiodic waveforms can be well explained by the propagating acoustic waves between the stalled shock and the rotating proto-neutron star surface, which suggests the appearance of the standing-accretion-shock instability. Although the GW signals exhibit strong variability between the two polarizations and different observer directions, they are within the detection limits of next-generation detectors such as KAGRA and Advanced LIGO, if the source with sufficient angular momentum is located in our Galaxy.

General relativistic neutrino transport using spectral methods

We present a new code, Lorene's Ghost (for Lorene's gravitational handling of spectral transport) developed to treat the problem of neutrino transport in supernovae with the use of spectral methods. First, we derive the expression for the nonrelativistic Liouville operator in doubly spherical coordinates (r, θ, ϕ, ϵ, Θ, Φ), and further its general relativistic counterpart. We use the 3 + 1 formalism with the conformally flat approximation for the spatial metric, to express the Liouville operator in the Eulerian frame. Our formulation does not use any approximations when dealing with the angular arguments (θ, ϕ, Θ, Φ), and is fully energy-dependent. This approach is implemented in a spherical shell, using either Chebyshev polynomials or Fourier series as decomposition bases. It is here restricted to simplified collision terms (isoenergetic scattering) and to the case of a static fluid. We finish this paper by presenting test results using basic configurations, including general relativistic ones in the Schwarzschild metric, in order to demonstrate the convergence properties, the conservation of particle number and correct treatment of some general relativistic effects of our code. The use of spectral methods enables to run our test cases in a six-dimensional setting on a single processor.