Divergence Correction Techniques for Maxwell Solvers Based on a Hyperbolic Model

On progressive blast envelope evolution of charged particles in electromagnetic fields

Представлен новый метод решения релятивистских уравнений движения заряженных частиц в электромагнитных полях, учитывающий условие постоянства их значений на каждом временном шаге. Проведено сравнение точности и эффективности вычислений при решении тестовых задач в дву- и трехмерной постановках на основе нового метода, метода Бориса и его модификаций. В каждом случае рассматривались варианты аналитически и дискретно заданных значений электрического и магнитного полей. At present, the Boris method is mostly common for the numerical solution of problems of the dynamics of charged particles in electromagnetic fields. In recent years, new modifications of the Boris method have appeared that are capable to simplify, refine, or speed up calculations. However, despite the rather large number of publications which consider algorithms for calculating the trajectories of charged particles, new, more accurate and high-performance algorithms are desired. The article presents a detailed analysis of some explicit methods for solving the problem of the motion of charged particles in electromagnetic fields, which differ in how they set the average values of velocities. A new scheme based on an analytical solution, which has not been studied before, is proposed. For the Boris, Vay, Higuera – Cary schemes and the new VD1 scheme, a comparative analysis of the accuracy, convergence and counting time for the relativistic and nonrelativistic cases is carried out. In the nonrelativistic case, the new scheme allows accurate solving of the equations for motion of charged particles, and in the relativistic case, the accuracy of the new scheme remains higher compared to other schemes. As the relativistic factor increases, the accuracy of all considered schemes for calculating the trajectory of particles in an electromagnetic field decreases. For large values of the relativistic parameter, the new scheme is more accurate than the other schemes and retains the second order. In practical calculations, when particles can have different speeds, the scheme automatically adjusts to the given speed of the particle. The presented new scheme for calculating trajectories can be important for solving a wide range of problems in astrophysics and thermonuclear fusion, where high accuracy in determining particle trajectories in inhomogeneous fields for large moments of time is required.

In this chapter we consider some problems in solid-state physics and in the theory of motion of charged particles in electromagnetic fields, whose solutions lead to incomplete cylindrical functions.

The most obvious components of particle accelerators and beam transport systems are those that provide the beam guidance system. Whatever the application may be, a beam of charged particles is designed to follow closely a predescribed path along a desired beam transport line or along a closed orbit in case of circular accelerators. The forces required to bend and direct the charged particle beam or to hold particles close to the ideal path are known as the Lorentz forces and are derived from electric and magnetic fields through the Lorentz equation.

In the papers (1.4) the exact expressions for the Green's functions of the Klein-Gordon and Dirae equations in some special types of external electromagnetic fields (namely, various combinations of uniform electric and magnetic fields and the fields of plane waves) were obtained by different and sometimes rather complicated methods. The possibility to obtain exact solutions in all these cases is explained by the fact that the corresponding relativistic problems can be reduced to solving the nonrelativistic SchrSdinger equations with quadratic Hamiltonians. Nonrelativistic systems with arbitrary quadratic N-dimensional Hamiltonians were recently studied in detail in the papers (5,e). Therefore in the present article we want to apply the general methods of ref. (5,e) to the relativistic equations in the external fields of the types described above and to show that these methods enable one to obtain exact Green's functions in all mentioned cases by means of avery simple procedure. Moreover, we shall obtain also exact results in some new interesting cases. For simplicity and brevity we consider only the Klein-Gordon equation. The (second order) Dirae equation can be considered analogously, but the formulae in this case are more cumbersome. We use the usual notations goo = -- gll ~ -- g22 = -- ga3 = l, t~ = c = 1, Pu = i~/~r~, # = 0, 1, 2, 3, where the eontravariant four-vector r is (t, x, y, z). The causal Green's function of the Klein-Gordon equation can be represented, according to (L2), as the integral over the proper time:

A plasma consists of charged particles moving relatively freely, but interacting through the electric and magnetic fields produced by each particle. It is well known that all except a very small (but nevertheless important) fraction of the detectable universe is a plasma. Most plasma physics can be expressed in terms of five fairly straightforward equations, the four Maxwell equations for electromagnetism and the equation of motion for charged particles in electromagnetic fields. Why then is plasma physics such a mathematically difficult subject and why is numerical simulation so important? The most basic reason is that any experimental plasma contains a very large number of particles, 1018–1020 in a typical fusion experiment. Moreover, particles in plasmas have a tendancy to act independently of each other. In conventional hydrodynamics, large numbers of particles can be described by a few simple numbers such as their temperature, density and mean velocity. Sometimes we can do this with a plasma but even then we have a few more numbers to include such as current and charge density, along with the electromagnetic fields. More usually, plasmas are not so simply described. Plasmas are not usually Maxwellian and we have to think in terms of distribution functions in phase space. Whenever we can, we model the non—Maxwellian nature of the plasma by transport models, e.g. for heat flow, but this is a specialised skill in itself. When one considers that conventional hydrodynamics is very complex and involves advanced numerical simulation, it is hardly surprising that simulation plays such an important role in plasma physics.

This note is concerned with an optimal control problem governed by the relativistic Maxwell--Newton--Lorentz equations, which describe the motion of charged particles in electro-magnetic fields and consists of a hyperbolic PDE system coupled with a nonlinear ODE. An external magnetic field acts as control variable. Additional control constraints are incorporated by introducing a scalar magnetic potential which leads to an additional state equation in the form of a very weak elliptic PDE. Existence and uniqueness for the state equation is shown and the existence of a global optimal control is established. Moreover, first-order necessary optimality conditions in the form of Karush--Kuhn--Tucker conditions are derived. A numerical test illustrates the theoretical findings.

An improved Boris algorithm for simulating the motion of charged particles in electromagnetic fields has been developed. This enhancement addresses the issue of inaccurate fast-scale cyclotron phase calculations present in the original Boris algorithm, while preserving its advantage in simulating slow-scale guiding center motion. As a result, it strikes a balance between low and high-frequency dynamics, overcoming the limitations of traditional second-order volume-preserving algorithms (VPAs) which are constrained to a single characteristic frequency. Test particle simulations indicate that, in most cases, the improved Boris algorithm achieves significantly higher accuracy than conventional VPAs when simulating cases involving various frequencies of electric field within a typical tokamak magnetic field, highlighting its superior efficacy in handling problems across a wide range of characteristic frequencies.

Charged Particles in Electromagnetic Fields

In this chapter we shall consider the motion of a charged particle in various types of electromagnetic field. We shall assume that the force which acts on the particle is the Lorentz force $$F = q[E + (v \times B)]$$ (7.1) in our usual notation. We shall also assume that the fields E and B are given, and thus we ignore for the moment the very difficult problem of how the fields produced by the particle itself affect its motion. We shall return to this much more difficult problem in Chapter 9.

The direct implicit particle-in-cell is a powerful kinetic method for researching plasma characteristics. However, it is time-consuming to obtain the future electromagnetic field in such a method since the field equations contain time-dependent matrix coefficients. In this work, we propose to explicitly push particles and obtain the future electromagnetic field based on the information about the particles in the future. The new method retains the form of implicit particle pusher, but the future field is obtained by solving the traditional explicit equation. Several numerical experiments, including the motion of charged particle in electromagnetic field, plasma sheath, and free diffusion of plasma into vacuum, are implemented to evaluate the performance of the method. The results demonstrate that the proposed method can suppress finite-grid-instability resulting from the coarse spatial resolution in electron Debye length through the strong damping of high-frequency plasma oscillation, while accurately describe low-frequency plasma phenomena, with the price of losing the numerical stability at large time-step. We believe that this work is helpful for people to research the bounded plasma by using particle-in-cell simulations.

Charged Particles in Electromagnetic Fields

Basic properties of plasmas are introduced, which are valid for an extremely wide range of plasma parameters. Plasmas are classified by different physical behaviour. The motion of charged particles in electromagnetic fields is revised with respect to drift motions. Adiabatic invariants are discussed and the kinetic description of plasmas is briefly presented. This Chapter is also a guideline connecting the subsequent Chapters.

For simple electromagnetic models of a rod and a clock, a change of shape of a rod and of the rate of clock when they are set in uniform motion is calculated exactly, employing the correct equation of motion of a charged particle in electromagnetic field and the universal boostability assumption. Thus it is demonstrated that, for the simple system considered, the length contraction and clock retardation can be interpreted as dynamical cause-and-effect phenomena, and not as kinematical effects as usually construed in conventional presentations of Special Relativity. It is argued that the perspective relativistic change of an object (corresponding to observations from two inertial frames), while certainly being an a causal effect, has a dynamical content in the sense that it is tantamount to an actual dynamical change of the object in one frame.

Direct observations of compact objects, in the form of radiation spectra, gravitational waves from VIRGO/LIGO, and forthcoming direct imaging, are currently one of the primary source of information on the physics of plasmas in extreme astrophysical environments. The modeling of such physical phenomena requires numerical methods that allow for the simulation of microscopic plasma dynamics in presence of both strong gravity and electromagnetic fields. In Bacchini et al. (2018) we presented a detailed study on numerical techniques for the integration of free geodesic motion. Here we extend the study by introducing electromagnetic forces in the simulation of charged particles in curved spacetimes. We extend the Hamiltonian energy-conserving method presented in Bacchini et al. (2018) to include the Lorentz force and we test its performance compared to that of standard explicit Runge-Kutta and implicit midpoint rule schemes against analytic solutions. Then, we show the application of the numerical schemes to the integration of test particle trajectories in general relativistic magnetohydrodynamic (GRMHD) simulations, by modifying the algorithms to handle grid-based electromagnetic fields. We test this approach by simulating ensembles of charged particles in a static GRMHD configuration obtained with the Black Hole Accretion Code (BHAC).