Vlasov-Maxwell System Research Articles

Quantum algorithm for the Vlasov equation

The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as $\mathcal{O}(N_v t)$ for a velocity grid with $N_v$ grid points and simulation time $t$, our quantum algorithm scales as $\mathcal{O}(\text{polylog}(N_v) t/\delta)$ where $\delta$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/\delta$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.

Orb5: A global electromagnetic gyrokinetic code using the PIC approach in toroidal geometry

This paper presents the current state of the global gyrokinetic code Orb5 as an update of the previous reference (Jolliet et al., 2007). The Orb5 code solves the electromagnetic Vlasov–Maxwell system of equations using a PIC scheme and also includes collisions and strong flows. The code assumes multiple gyrokinetic ion species at all wavelengths for the polarization density and drift-kinetic electrons. Variants of the physical model can be selected for electrons such as assuming an adiabatic response or a “hybrid” model in which passing electrons are assumed adiabatic and trapped electrons are drift-kinetic. A Fourier filter as well as various control variates and noise reduction techniques enable simulations with good signal-to-noise ratios at a limited numerical cost. They are completed with different momentum and zonal flow-conserving heat sources allowing for temperature-gradient and flux-driven simulations. The code, which runs on both CPUs and GPUs, is well benchmarked against other similar codes and analytical predictions, and shows good scalability up to thousands of nodes.

Effect of external magnetic field on wakefield generation in underdense plasma slab using backward semi-Lagrangian Vlasov code

Laser wakefield excitation in the interaction of femtosecond laser pulse with an underdense thin magnetized plasma slab is studied by one dimensional relativistic Vlasov-Maxwell system of equations. The interaction is modeled by relativistic Vlasov equation in propagation direction of laser pulse and fluid equations in transverse direction. Backward semi-Lagrangian method (BSL) is used for numerical solution of Vlasov equation. Generation of wakefields by a right and left handed circularly polarized (RCP) and (LCP) laser pulses in the interaction with nonmagnetized and magnetized plasma slab are examined and compared. Two directions are considered for external magnetic field, parallel and antiparallel to direction of laser pulse propagation. The results show that applying magnetic field along (opposite to) the propagation direction of RCP (LCP) laser pulse increases amplitude of density steepening and consequently wakefields amplitude, and mean kinetic energy of electrons compared with nonmagnetized plasma. The results are in complete agreement with previous analytical and numerical reports.

Convergence of hp-Streamline Diffusion Method for Vlasov–Maxwell System

In this paper we study stability and convergence for hp-streamline diffusion (SD) finite element method for the, relativistic, time-dependent Vlasov–Maxwell (VM) system. We consider spatial domain and velocities The objective is to show globally optimal a priori error bound of order for the SD approximation of the VM system; where is the mesh size and is the spectral order. Our estimates rely on the local version with hK being the diameter of the phase-space-time element K and pK the spectral order for K. The optimal hp estimates require an exact solution in the Sobolev space Numerical implementations, performed for examples in one space- and two velocity dimensions, are justifying the robustness of the theoretical results.

A low-rank projector-splitting integrator for the Vlasov–Maxwell equations with divergence correction

The Vlasov–Maxwell equations are used for the kinetic description of magnetized plasmas. As they are posed in an up to 3+3 dimensional phase space, solving this problem is extremely expensive from a computational point of view. In this paper, we exploit the low-rank structure in the solution of the Vlasov equation. More specifically, we consider the Vlasov–Maxwell system and propose a dynamic low-rank integrator. The key idea is to approximate the dynamics of the system by constraining it to a low-rank manifold. This is accomplished by a projection onto the tangent space. There, the dynamics is represented by the low-rank factors, which are determined by solving lower-dimensional partial differential equations. The proposed scheme performs well in numerical experiments and succeeds in capturing the main features of the plasma dynamics. We demonstrate this good behavior for a range of test problems. The coupling of the Vlasov equation with the Maxwell system, however, introduces additional challenges. In particular, the divergence of the electric field resulting from Maxwell's equations is not consistent with the charge density computed from the Vlasov equation. We propose a correction based on Lagrange multipliers which enforces Gauss' law up to machine precision.

Distinct features of Alfvén wave in non-extensive plasmas

Since the time of its discovery, Alfvén wave has been the subject of attention in both theoretical and experimental plasma physics. To date, many different aspects of this wave have been studied using various models. In this study, we adopt linearized Vlasov–Maxwell system and investigate very important characteristics of the wave: electromagnetic field perturbation and Poynting flux, which gives a measure of the electromagnetic energy stored in the field. The plasma which supports the wave is considered to be non-extensive. It is found that the electromagnetic field behaves totally different from the Maxwellian case, and so does the resultant transport of energy. Due to the wide spectrum of the non-extensive parameter q, the results can incorporate a broad range of possibilities which suggest that the energy transport of the wave over short and long distances may somehow be controlled by the non-extensivity of the plasmas.

Onsager-type conjecture and renormalized solutions for the relativistic Vlasov–Maxwell system

In this paper we give a proof of an Onsager type conjecture on conservation of energy and entropies of weak solutions to the relativistic Vlasov--Maxwell equations. As concerns the regularity of weak solutions, say in Sobolev spaces $W^{\alpha,p}$, we determine Onsager type exponents $\alpha$ that guarantee the conservation of all entropies. In particular, the Onsager exponent $\alpha$ is smaller than $\alpha = 1/3$ established for fluid models. Entropies conservation is equivalent to the renormalization property, which have been introduced by DiPerna--Lions for studying well-posedness of passive transport equations and collisionless kinetic equations. For smooth solutions renormalization property or entropies conservation are simply the consequence of the chain rule. For weak solutions the use of the chain rule is not always justified. Then arises the question about the minimal regularity needed for weak solutions to guarantee such properties. In the DiPerna--Lions and Bouchut--Ambrosio theories, renormalization property holds under sufficient conditions in terms of the regularity of the advection field, which are roughly speaking an entire derivative in some Lebesgue spaces (DiPerna--Lions) or an entire derivative in the space of measures with finite total variation (Bouchut--Ambrosio). In return there is no smoothness requirement for the advected density, except some natural a priori bounds. Here we show that the renormalization property holds for an electromagnetic field with only a fractional space derivative in some Lebesgue spaces. To compensate this loss of derivative for the electromagnetic field, the distribution function requires an additional smoothness, typically fractional Sobolev differentiability in phase-space. As concerns the conservation of total energy, if the macroscopic kinetic energy is in $L^2$, then total energy is preserved.

Solving the Vlasov–Maxwell equations using Hamiltonian splitting

In this paper, the numerical discretizations based on Hamiltonian splitting for solving the Vlasov–Maxwell system are constructed. We reformulate the Vlasov–Maxwell system in Morrison–Marsden–Weinstein Poisson bracket form. Then the Hamiltonian of this system is split into five parts, with which five corresponding Hamiltonian subsystems are obtained. The splitting method in time is derived by composing the solutions to these five subsystems. Combining the splitting method in time with the Fourier spectral method and finite volume method in space gives the full numerical discretizations which possess good conservation for the conserved quantities including energy, momentum, charge, etc. In numerical experiments, we simulate the Landau damping, Weibel instability and Bernstein wave to verify the numerical algorithms.

Normal-mode-based theory of collisionless plasma waves

The Van Kampen normal-mode method is applied in a comprehensive study of the linear wave perturbations of a homogeneous, magnetized and finite-temperature plasma, described by the collisionless Vlasov–Maxwell system in its non-relativistic version. The analysis considers a stable, Maxwellian background, but is otherwise completely general in that it allows for arbitrary wave propagation direction relative to the equilibrium magnetic field, multiple plasma species and general polarization states of the perturbed electromagnetic fields. A convenient formulation is introduced whereby the generator of the time advance is a Hermitian operator, analogous to the Hamiltonian in the Schrödinger equation of quantum mechanics. This guarantees a real frequency spectrum and complete bases of normal modes. Expansions in these normal-mode bases yield immediately the solutions of initial-value problems for general initial conditions. With standard initial conditions and propagation direction parallel to the equilibrium magnetic field, all the familiar results obtained following Landau’s Laplace transform approach are recovered. Considering such parallel propagation, the present work shows also explicitly and provides an example of how to construct special initial conditions that result in different, damped but otherwise arbitrarily prescribed time variations of the macroscopic variables. The known dispersion relations for perpendicular propagation are also recovered.

Combining particle-in-cell and direct simulation Monte Carlo for the simulation of reactive plasma flows

A combined approach for the simulation of reactive, neutral, partially or fully ionized plasma flows is presented. This is realized in a code framework named “PICLas” for the approximate solution of the Boltzmann equation by particle based methods. PICLas combines the particle-in-cell method for the collisionless Vlasov–Maxwell system and the direct simulation Monte Carlo method for neutral reactive flows. Basic physical and mathematical modeling of both methods is addressed, and some application examples are presented in order to demonstrate the capabilities and the broad applicability of the solution strategy.

Numerical solution of the Vlasov-Maxwell system of equations for cylindrical plasmas

We solved numerically the Vlasov-Maxwell system of equations for a bounded cylindrical and radial inhomogeneous plasma which is confined by a strong magnetic field directed along the axis cylinder. Through this solution we found numerically the radial structure of the axial electric field corresponding to the high frequency fundamental transverse magnetic mode propagating in the cylindrical wave guide. Our result shows that the intensity of the electric field tends to be higher in those regions where the plasma is denser and also the field presents oscillations with intensities that decrease and vanish at the radial plasma boundary. This behavior could be relevant in the design of efficient modern plasma based particle accelerators that use the axial electric field to achieve this task.

Sparse grid discontinuous Galerkin methods for the Vlasov-Maxwell system

In this paper, we develop sparse grid discontinuous Galerkin (DG) schemes for the Vlasov-Maxwell (VM) equations. The VM system is a fundamental kinetic model in plasma physics, and its numerical computations are quite demanding, due to its intrinsic high-dimensionality and the need to retain many properties of the physical solutions. To break the curse of dimensionality, we consider the sparse grid DG methods that were recently developed in [20,21] for transport equations. Such methods are based on multiwavelets on tensorized nested grids and can significantly reduce the numbers of degrees of freedom. We formulate two versions of the schemes: sparse grid DG and adaptive sparse grid DG methods for the VM system. Their key properties and implementation details are discussed. Accuracy and robustness are demonstrated by numerical tests, with emphasis on comparison of the performance of the two methods, as well as with their full grid counterparts.

Vlasov-Maxwell system with physical reasonableness constraints

Complete mathematical description on plasmas contains not only some coupled partial differential equations (PDEs) reflecting physics laws but also some inequalities reflecting physical reasonableness requirement. The phrase ‘physical reasonableness’ refers to that solutions should always correspond to non-negative-valued probability distribution function and non-negative-valued particle density. This work displays a universal strict method on the V-M system with a constraint inequality f ≥ 0∀ r, p, t. It treats the Vlasov equation as a recurrence formula relating expansion coefficient functions of the power series solution of the equation, and strictly demonstrates that the constraint determines the shape, or geometric characteristics, of space-time contours of the fluid velocity field. Consequently, the constraint inequality makes those PDEs reflecting physics laws to be expressed finally as an ordinary equation of the fluid velocity field with respect to its contour.

Linearized Coulomb Collision Operator for Simulation of Interpenetrating Plasma Streams

We present an extension of a linearized Coulomb collision operator, previously used in several Eulerian kinetic codes for like-species collisions and unlike-species collisions in the case where the backgrounds about which the linearization is made all are in collisional equilibrium, to the situation of interpenetrating plasma streams. In the latter case, the backgrounds cannot be taken to be in equilibrium and a significant generalization is required. Our development is targeted toward the Eulerian kinetic plasma code LOKI, which evolves the Vlasov–Poisson or Vlasov–Maxwell system in a Cartesian “2 + 2-dimensional” phase space. The extended operator has been implemented in a test code, and results of both quantitative verification and qualitative “realizability” tests are presented.

The relativistic Vlasov–Maxwell equations for strongly magnetized plasmas

An important challenge in plasma physics is to determine whether ionized gases can be confined by strong magnetic fields. After properly formulating the model, this question leads to a penalized version of the Relativistic Vlasov Maxwell system, marked by the role of a singular factor e −1 corresponding to the inverse of a cyclotron frequency. In this paper, we prove in this context the existence of classical C 1-solutions for a time independent of e. We also investigate the stability of these smooth solutions.

Fourier–Hermite decomposition of the collisional Vlasov–Maxwell system: implications for the velocity-space cascade

Turbulence at kinetic scales is an unresolved and ubiquitous phenomenon that characterizes both space and laboratory plasmas. Recently, new theories, in situ spacecraft observations and numerical simulations suggest a novel scenario for turbulence, characterized by a so-called phase-space cascade—the formation of fine structures, both in physical and velocity-space. This new concept is here extended by directly taking into account the role of inter-particle collisions, modeled through the nonlinear Landau operator or the simplified Dougherty operator. The characteristic times, associated with inter-particle correlations, are derived in the above cases. The implications of introducing collisions on the phase-space cascade are finally discussed.

Why the particle-in-cell method captures instability enhanced collisions

While the particle-in-cell (PIC) method has been the subject of years of theoretical study, the common misconception that PIC approximates the collisionless Vlasov-Maxwell system of equations, or the Vlasov-Poisson system for the electrostatic case, is widely cited. In this paper, a PIC-relevant generalization of the instability enhanced Lenard-Balescu collision operator is derived. The analysis of this collision operator demonstrates that while coulomb collisions are truncated by the electric field grid resolution, longer range collisions that arise from imperfect shielding in the presence of instabilities persist. These instability enhanced collisions are completely captured by PIC as long as there is sufficient grid resolution to resolve the wavenumber of the unstable modes of the instability. The inclusion of this behavior is akin to particle wave interactions in the Vlasov based quasilinear theory but with one important difference. Quasilinear theory requires an initial spectral energy density which cannot be supplied self-consistently from within the theory because its initial value is determined by non-Vlasov collisional effects. With the proper electric field grid resolution, the initial spectral energy density is included self-consistently, along with the generation of plasma waves originating from discrete particle motion. Predictions of the grid resolution effect are found to be in agreement with PIC simulations at varying grid resolutions.

Regularity of weak solutions for the relativistic Vlasov–Maxwell system

We investigate the regularity of weak solutions of the relativistic Vlasov–Maxwell system by using Fourier analysis and the smoothing effect of low velocity particles. This smoothing effect has been used by several authors (see Glassey and Strauss 1986; Klainerman and Staffilani, 2002) for proving existence and uniqueness of [Formula: see text]-regular solutions of the Vlasov–Maxwell system. This smoothing mechanism has also been used to study the regularity of solutions for a kinetic transport equation coupled with a wave equation (see Bouchut, Golse and Pallard 2004). Under the same assumptions as in the paper “Nonresonant smoothing for coupled wave[Formula: see text]+[Formula: see text]transport equations and the Vlasov–Maxwell system”, Rev. Mat. Iberoamericana 20 (2004) 865–892, by Bouchut, Golse and Pallard, we prove a slightly better regularity for the electromagnetic field than the one showed in the latter paper. Namely, we prove that the electromagnetic field belongs to [Formula: see text], with [Formula: see text].

Quadratic conservative scheme for relativistic Vlasov–Maxwell system

For more than half a century, most of the plasma scientists have encountered a violation of the conservation laws of charge, momentum, and energy whenever they have numerically solved the first-principle equations of kinetic plasmas, such as the relativistic Vlasov–Maxwell system. This fatal problem is brought by the fact that both the Vlasov and Maxwell equations are indirectly associated with the conservation laws by means of some mathematical manipulations. Here we propose a quadratic conservative scheme, which can strictly maintain the conservation laws by discretizing the relativistic Vlasov–Maxwell system. A discrete product rule and summation-by-parts are the key players in the construction of the quadratic conservative scheme. Numerical experiments of the relativistic two-stream instability and relativistic Weibel instability prove the validity of our computational theory, and the proposed strategy will open the doors to the first-principle studies of mesoscopic and macroscopic plasma physics.

Reformulation of Vlasov-Maxwell system and a new method for its numerical solution

Reformulation of Vlasov-Maxwell system and a new method for its numerical solution