An efficient energy-preserving algorithm for the Lorentz force system
Introduction
Geometric numerical integration methods have been an area of great interest and active research in the recent decades. A numerical method is called geometric method if it can conserve the geometric properties of a system up to round-off error [11], [12], [13]. Geometric methods, such as symplectic methods, symmetric methods, volume-preserving methods, energy-preserving methods and so on, have been designed and analysed in many application areas [10], [19], [20], [25], [29], [38], [39].
The motion of charged particles under the influence of electromagnetic fields is a fundamental process in the collective dynamics of magnetized plasmas [2]. For the single particle motion which satisfies the Lorentz force equations, the system can be rewritten as a Hamiltonian formulation [27], [31]. Long-term numerical simulations on trajectories of charged particles have been widely used to study their dynamical behaviours. While non-geometric methods such as the standard 4th order Runge-Kutta method cannot trace the trajectory accurately after a long time of computation due to the error accumulation. In contrast, great advances have been achieved in long-term accurate simulations of charged particle dynamics with the application of geometric integration methods, for instance, the volume-preserving algorithms [21], [45], the Boris method which can also conserve phase space volume [3], [4], [34], [37], the variational symplectic method [28], [32], [33] and so on. Obviously, energy-preserving is one of the most relevant features characterizing a Hamiltonian system. In 2016, a similar energy-preserving BDLI method for the system is proposed [26]. Based on the Boole’s rule, the non-polynomial Hamiltonian can be preserved up to the round-off error. However, there is no exactly energy-preserving method for the Lorentz force system.
Various energy-preserving methods have been proposed since several decades [14], [17], [30], [35], [36]. It is known that discrete gradient methods are among the most popular candidates for constructing integral preserving schemes for ordinary differential equations. This was first investigated by Gonzalez [17] in 1996. Later, Matsuo [30] proposed a discrete variational method for nonlinear wave equation [30]. The averaged vector field method which is a B-series method has been proposed [14], [36]. Brugnano and Iavernaro derived the discrete line integral (DLI) methods [22], [23], the Hamiltonian boundary value methods [7], [24] and the line integral methods [5], [6], [8], [9]. In addition, based on the original ideal of Baida et al. [1] and Guillen et al. [18] in the treatment of liquid crystal models, Yang et al. proposed the (invariant) energy quadratization (EQ or IEQ) method for thermodynamics as hydrodynamic models [15], [16], [40], [41], [42], [43], [44], [46]. Applying the IEQ strategy for Hamiltonian system to obtain a efficient and exactly energy-preserving algorithm is a interesting and meaningful topic.
In this paper, the Lorentz force system is written as a Hamiltonian system. We transform the original Hamiltonian energy functional into a quadratic term by introducing an auxiliary variable. Then, applying the Crank-Nicolson scheme with the Richardson extrapolation technique for the new system, an exactly energy-preserving method is obtained. The new scheme is a linear-implicit one for canonical Hamiltonian system, so we only need to solve a linear system at each time step, which can efficiently simulate the motion of charged particles in constant magnetic field.
The paper is organized as follows: In Section 2, the dynamics of charged particles in the electromagnetic field is shown and it is written as a Hamiltonian system. In Section 3, we use the IEQ approach to solve the Hamiltonian system and a new energy-preserving method for the Lorentz force system is obtained. Numerical experiments are presented in Section 4 to confirm the theoretical results of the new formula in comparison with the well-known Boris method [4] and the energy-preserving BDLI method [26]. We finish the paper with conclusions in Section 5.
Section snippets
Hamiltonian form of the Lorentz force system
In this section, we review the Hamiltonian form of the Lorentz force system [21], [27], [31]. For a charged particle in the electromagnetic field, its dynamics is governed by the Newton-Lorentz equationwhere x is the position of the charged particle, m is the mass, and q denotes the electric charge. For convenience, here we assume that and are static, thus and with and φ the potentials.
Let the conjugate momentum be then the system (2.1) is
Invariant energy quadratization (IEQ) approach for the Lorentz force system
In this section, we apply the IEQ approach to solve the Hamiltonian system (2.5). Firstly, we introduce some notations:Assuming φ is bounded from below, let where and we choose suitable constant C0 such that H2(z) > 0. Then we introduce a Lagrange multiplier (auxiliary variable) and rewrite (2.5) asInserting (3.1) into (3.2), we can conclude that the
Numerical experiments
In this section, we numerically test the EPIEQ method (3.3) and (3.4). The results are compared with the well-known Boris method [4] and the energy-preserving BDLI method [26].
Conclusions
The single-particle motion which satisfies the Lorentz force system plays an important role in plasmas and the energy is one of the most relevant features characterizing the system. However, there is few numerical methods which can exactly conserve the energy of the system. In this paper, we apply the IEQ approach to solve the Lorentz force system and a new exactly energy-preserving EPIEQ method is proposed. The EPIEQ method will become a linear-implicit scheme if the magnetic field is constant
Acknowledgments
The authors would like to express sincere gratitude to the reviewers for their constructive suggestions which helped to improve the quality of this paper. This work is supported by the Fundamental Research Funds for the Central Universities and the Innovation Foundation of BUPT for Youth.
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