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Integrating batched sparse iterative solvers for the collision operator in fusion plasma simulations on GPUs
Not Available
Spectrum of the Boltzmann collision operator for λϕ4 theory in the classical regime
Not Available
Machine Learning for Accelerating Direct-Simulation Monte-Carlo Collision Operators.
Abstract not provided.
Neural-Network Based Collision Operators for the Boltzmann Equation.
Abstract not provided.
Machine Learning for Accelerating Direct-Simulation Monte-Carlo Collision Operators.
Abstract not provided.
Driven electrostatic phase space vortices in a 1D weakly dissipative Vlasov–Poisson system
In this paper, the effect of collisions on driven electrostatic phase space vortices is analyzed by means of Eulerian simulation for two different collision models. It was demonstrated recently that in the absence of collisions, at late times, steady state phase space vortices manifest to form a plateau in the resonant region of the particle velocity distribution function, due to trapping of particles supporting multiextrema giant phase space vortices (PSVs). In the presence of collisions, over long time, this multiextrema plateau are found to smooth out, since collisions drive the velocity distribution toward Maxwellian, irrespective of how weak the collisions are as long as they are non-zero. In these conditions, kinetic processes and collisionality are found to be in competition, and the evolution of the plasma is found, therefore, to be a result of nontrivial combination of these two effects. An attempt has been made by means of numerical simulations to study the effect of weak collisionality on the electrostatic driven phase space vortices with two types of collision operators: (1) Bhatnagar–Gross–Krook (Krook) collision operator, where the colliding particles can be treated as isolated pairs and, (2) Fokker–Planck (FP) type collision operator (Zakharov–Karpman) in one dimension, where many weak collisions lead to particle diffusion in velocity space. It is shown that depending on the collision model used, the nature of smoothing in velocity space of giant PSVs results in qualitatively very different phase space structures. However, irrespective of the collision model used, excess density fractions over 10% are retained.
Krash-ESL-Final report
During the period of the current Grant, the effort of the UCSD researchers was focused on: i) The application of the gyrokinetic code COGENT code to study the kinetic drift wave instability and its impact on blob dynamics, and ii) The modeling of edge plasma transport in collisional and semi-collisional regimes, and iii) Impurity transport in edge plasma and the incorporation of plasma-impurity collision operator (hereafter unlike particle collision operator) into COGENT. Over the years four UCSD graduate students (J. Angus, W. Lee, Y. Zhang, and A. Knyazev), funded under this Grant, contributed to the COGENT development/utilization and defended their PhD theses, which are based, partly, on their research related to the COGENT. Three of them (J. Angus, W. Lee, and Y. Zhang) have received “The Best MAE Graduate Student of the Year” awards. Whereas W. Lee after defending his PhD thesis returned back to his home country (South Korea), J. Angus, and Y. Zhang are continuing their research in plasma physics in the LLNL and LANL, whereas A. Knyazev currently is a postdoctoral fellow at Columbia University working on the theoretical aspects of advanced stellarators. The results obtained in our studies were disseminated in many papers published in first-class refereed scientific journals and presented on prestigious scientific meetings including the Annual APS DPP Meeting, Sherwood Fusion Theory Conference, IAEA Fusion Energy Conferences, International Conferences on Plasma-Surface Interactions in Controlled Fusion Devices, International Workshops on Edge Plasma Theory in Fusion Devices, etc. In what follows we highlight the major accomplishments of the UCSD Team in the main areas of our research.
Asymptotic-preserving dynamical low-rank method for the stiff nonlinear Boltzmann equation
In kinetic theory, numerically solving the full Boltzmann equation is extremely expensive. This is because the Boltzmann collision operator involves a high-dimensional, nonlinear integral that must be evaluated at each spatial grid point and every time step. The challenge becomes even more pronounced in the fluid (strong collisionality) regime, where the collision operator exhibits strong stiffness, causing explicit time integrators to impose severe stability restrictions. In this paper, we propose addressing this problem through a dynamical low-rank (DLR) approximation. The resulting algorithm requires evaluating the Boltzmann collision operator only r 2 times, where r, the rank of the approximation, is much smaller than the number of spatial grid points. We propose a novel DLR integrator, called the XL integrator, which reduces the number of steps compared to the available alternatives (such as the projector splitting or basis update & Galerkin (BUG) integrator). For a class of problems including the Boltzmann collision operator which enjoys a separation property between physical and velocity space, we further propose a specialized version of the XL integrator, called the sXL integrator. This version requires solving only one differential equation to update the low-rank factors. Furthermore, the proposed low-rank schemes are asymptotic-preserving, meaning they can capture the asymptotic fluid limit in the case of strong collisionality. Our numerical experiments demonstrate the efficiency and accuracy of the proposed methods across a wide range of regimes, from non-stiff (kinetic) to stiff (fluid).
Semi-Lagrangian nodal discontinuous Galerkin method for the BGK model
In this work, we propose a semi-Lagrangian (SL) nodal discontinuous Galerkin (DG) solver for the BGK equation. The BGK model was introduced by Bhatnagar, Gross, and Krook [1] as a relaxation model for the fundamental Boltzmann equation [5], which describes the kinetic dynamic of rarefied gases with a probability distribution function. The challenges of designing efficient numerical schemes for the Boltzmann equation mainly come from its high dimensionality and complicated nonlinear collision operator. The BGK model gains interests since it has much lower computational cost, due to the relatively simple structure of the relaxation operator in replacement of the collision operator, while simultaneously preserving several important physical properties, such as macroscopic quantities and dissipation of entropy.
A generalized Boltzmann kinetic theory for strongly magnetized plasmas with application to friction
Coulomb collisions in plasmas are typically modeled using the Boltzmann collision operator, or its variants, which apply to weakly magnetized plasmas in which the typical gyroradius of particles significantly exceeds the Debye length. Conversely, O’Neil has developed a kinetic theory to treat plasmas that are so strongly magnetized that the typical gyroradius of particles is much smaller than the distance of the closest approach in a binary collision. Here, we develop a generalized collision operator that applies across the full range of magnetization strength. Since there is no closed-form solution for the scattering cross section when plasma is strongly magnetized, the input to the collision operator is obtained by numerically calculating particle trajectories. To demonstrate novel physics associated with strong magnetization, it is used to compute the friction force on a massive test charge. In addition to the traditional stopping power component, this is found to exhibit a transverse component that is perpendicular to both the velocity and Lorentz force vectors in the strongly magnetized regime, as was predicted recently using linear response theory. As a result, good agreement is found between the collision theory and linear response theory in the regime in which both apply, but the new collision theory also applies to stronger magnetization strength regimes than the linear response theory is expected to apply in.
Multi-species collisions for delta-f gyrokinetic simulations: Implementation and verification with GENE
we report a multi-species linearized collision operator based on the model developed by Sugama et al. has been implemented in the nonlinear gyrokinetic code, GENE. Such a model conserves particles, momentum, and energy to machine precision, and is shown to have negative definite free energy dissipation characteristics, satisfying Boltzmann’s H-theorem, including for realistic mass ratio. Finite Larmor Radius (FLR) effects have also been implemented into the local version of the code. For the global version of the code, the collision operator has been developed to allow for block-structured velocity space grids, allowing for computationally tractable collisional global simulations. The validity of the collision operator has been demonstrated by relaxation and conservation tests, as well as appropriate benchmarks. The newly implemented operator shall be used in future simulations to study magnetically confined fusion plasma turbulence and transport in more extreme regions with higher collisionality.
Multi-species collisions for delta-f gyrokinetic simulations: Implementation and verification with GENE
A multi-species linearized collision operator based on the model developed by Sugama et al. has been implemented in the nonlinear gyrokinetic code, GENE. Such a model conserves particles, momentum, and energy to machine precision, and is shown to have negative definite free energy dissipation characteristics, satisfying Boltzmann’s H-theorem, including for realistic mass ratio. Finite Larmor Radius (FLR) effects have also been implemented into the local version of the code. For the global version of the code, the collision operator has been developed to allow for block-structured velocity space grids, allowing for computationally tractable collisional global simulations. The validity of the collision operator has been demonstrated by relaxation and conservation tests, as well as appropriate benchmarks. The newly implemented operator shall be used in future simulations to study magnetically confined fusion plasma turbulence and transport in more extreme regions with higher collisionality.
Fluid-Kinetic Coupling: Advanced Discretizations for Simulations on Emerging Heterogeneous Architectures (LDRD FY20-0643)
Plasma physics simulations are vital for a host of Sandia mission concerns, for fundamental science, and for clean energy in the form of fusion power. Sandia's most mature plasma physics simulation capabilities come in the form of particle-in-cell (PIC) models and magnetohydrodynamics (MHD) models. MHD models for a plasma work well in denser plasma regimes when there is enough material that the plasma approximates a fluid. PIC models, on the other hand, work well in lower-density regimes, in which there is not too much to simulate; error in PIC scales as the square root of the number of particles, making high-accuracy simulations expensive. Real-world applications, however, almost always involve a transition region between the high-density regimes where MHD is appropriate, and the low-density regimes for PIC. In such a transition region, a direct discretization of Vlasov is appropriate. Such discretizations come with their own computational costs, however; the phase-space mesh for Vlasov can involve up to six dimensions (seven if time is included), and to apply appropriate homogeneous boundary conditions in velocity space requires meshing a substantial padding region to ensure that the distribution remains sufficiently close to zero at the velocity boundaries. Moreover, for collisional plasmas, the right-hand side of the Vlasov equation is a collision operator, which is non-local in velocity space, and which may dominate the cost of the Vlasov solver. The present LDRD project endeavors to develop modern, foundational tools for the development of continuum-kinetic Vlasov solvers, using the discontinuous Petrov-Galerkin (DPG) methodology, for discretization of Vlasov, and machine-learning (ML) models to enable efficient evaluation of collision operators. DPG affords several key advantages. First, it has a built-in, robust error indicator, allowing us to adapt the mesh in a very natural way, enabling a coarse velocity-space mesh near the homogeneous boundaries, and a fine mesh where the solution has fine features. Second, it is an inherently high-order, high-intensity method, requiring extra local computations to determine so-called optimal test functions, which makes it particularly suited to modern hardware in which floating-point throughput is increasing at a faster rate than memory bandwidth. Finally, DPG is a residual-minimizing method, which enables high-accuracy computation: in typical cases, the method delivers something very close to the $L^2$ projection of the exact solution. Meanwhile, the ML-based collision model we adopt affords a cost structure that scales as the square root of a standard direct evaluation. Moreover, we design our model to conserve mass, momentum, and energy by construction, and our approach to training is highly flexible, in that it can incorporate not only synthetic data from direct-simulation Monte Carlo (DSMC) codes, but also experimental data. We have developed two DPG formulations for Vlasov-Poisson: a time-marching, backward-Euler discretization and a space-time discretization. We have conducted a number of numerical experiments to verify the approach in a 1D1V setting. In this report, we detail these formulations and experiments. We also summarize some new theoretical results developed as part of this project (published as papers previously): some new analysis of DPG for the convection-reaction problem (of which the Vlasov equation is an instance), a new exponential integrator for DPG, and some numerical exploration of various DPG-based time-marching approaches to the heat equation. As part of this work, we have contributed extensively to the Camellia open-source library; we also describe the new capabilities and their usage. We have also developed a well-documented methodology for single-species collision operators, which we applied to argon and demonstrated with numerical experiments. We summarize those results here, as well as describing at a high level a design extending the methodology to multi-species operators. We have released a new open-source library, MLC, under a BSD license; we include a summary of its capabilities as well.
A comprehensive conductivity model for drift and micro-tearing modes
The parallel electrical conductivity is a crucial parameter in the study of the linear stability of drift-modes like the micro-tearing mode (MTM). The conductivity enters by closing the electromagnetic tearing layer equations. Recent progress in the understanding of the pedestal suggests that the MTM could play an important role in its structure and evolution. For this reason, we revisit and improve previous model conductivities. This parameter is generally derived from the linearized drift kinetic equation. In the past literature, it has been computed using either simplified collision operators or neglecting the spatial dependence away from the rational surface. Here, a fully consistent expression for the conductivity that would accurately model the pedestal has not been available. By applying a novel variational procedure and with the full Fokker Plank collision operator, including electron–electron collisions, we compute a closed expression for the parallel electrical conductivity in the form of a rational function.
A Particle Method for the Multispecies Landau Equation
Abstract The multispecies Landau collision operator describes the two-particle, small scattering angle or grazing collisions in a plasma made up of different species of particles such as electrons and ions. Recently, a structure preserving deterministic particle method (Carrillo et al. in J. Comput. Phys. 7:100066, 2020) has been developed for the single species spatially homogeneous Landau equation. This method relies on a regularization of the Landau collision operator so that an approximate solution, which is a linear combination of Dirac delta distributions, is well-defined. Based on a weak form of the regularized Landau equation, the time dependent locations of the Dirac delta functions satisfy a system of ordinary differential equations. In this work, we extend this particle method to the multispecies case, and examine its conservation of mass, momentum, and energy, and decay of entropy properties. We show that the equilibrium distribution of the regularized multispecies Landau equation is a Maxwellian distribution, and state a critical condition on the regularization parameters that guarantees a species independent equilibrium temperature. A convergence study comparing an exact multispecies Bobylev-Krook-Wu (BKW) solution to the particle solution shows approximately 2nd order accuracy. Important physical properties such as conservation, decay of entropy, and equilibrium distribution of the particle method are demonstrated with several numerical examples.
Electron kinetics in a high- Z plasmoid
The problem of the electron dynamics on a closed magnetic field line passing through a high-Z plasmoid is considered. The electron kinetic equation is integrated over bounce motion and pitch angle, reducing the independent variables to a single adiabatic invariant plus time. Integration of the full Landau self-collision operator is carried out exactly, resulting in a nonlinear integro-differential operator in the new invariant. Conservation laws and the H theorem of the integrated self-collision operator are proven. Numerical solutions of the integrated kinetic equation are obtained with a self-consistent quasineutral electric potential, given the initial condition of a cold plasmoid immersed in a hot ambient plasma. The fact that cold electrons are deeply trapped in a potential with a parabolic peak leads to exactly 3/4 the usual rate of collisional heating by the ambient plasma, independent of any other parameters.
Final Technical Report for the SciDAC Center for the Study of Plasma Microturbulence
This final technical report for the SciDAC Center for the Study of Plasma Microturbulence documents contributions from MIT to the project over the period of the award (February 15, 2011 to February 14, 2020). This project focused on the development of neoclassical and gyrokinetic simulation capabilities relevant for the description of particle, momentum and energy transport in tokamak fusion plasmas. Extensive validation against experimental results from Alcator C-Mod and DIII-D experiments was performed. Overall areas of emphasis included gyrokinetic simulations, experiments, and observation of density gradient driven Trapped Electron Mode (TEM) turbulence relevant to burning plasmas; theory and gyrokinetic simulations of the parametric variation of the nonlinear upshift of the TEM critical density gradient; identification of a new mechanism for the isotope effect due to density gradient driven TEM turbulence; new synthetic diagnostics based on full-wave simulations which enabled unprecedented agreement between gyrokinetic simulations and measured fluctuation spectra with and without strong electron heating; development of new continuum pedestal neoclassical and turbulence codes; formulation and implementation of the first gyrokinetic exact Fokker-Planck collision operator in gyrokinetic turbulence simulations; development of new spectral approaches in velocity space for highly efficient treatments of collisions; new model collision operators implemented in the discontinuous Galerkin gyrokinetic edge turbulence code GKEYLL; and a new model and spectral code to study the interaction of ion- and electron-scale turbulence, which will serve as an efficient test-bed for multi-rate and multi-scale methods. These fundamental advances contribute to the development of a validated, predictive capability for turbulent and collisional particle, momentum and energy transport in both the core and edge of magnetically confined fusion plasmas.