Efficient energy-preserving methods for charged-particle dynamics

Abstract

In this paper, energy-preserving methods are formulated and studied for solving chargedparticle dynamics. We first formulate the scheme of energy-preserving methods and analyze its basic properties including algebraic order and symmetry. Then it is shown that these novel methods can exactly preserve the energy of charged-particle dynamics. Moreover, the long time momentum conservation is studied along such energy-preserving methods. A numerical experiment is carried out to illustrate the notable superiority of the new methods in comparison with the popular Boris method in the literature.

Error estimates of some splitting schemes for charged-particle dynamics under strong magnetic field

“…It is noted that Algorithm 2.1 is implicit, while the nonlinear equation (2.4) is independent of ε. Therefore, compared with other implicit energy-preserving schemes [4,20,33,34,36] for solving CPD (1.1), the computational cost of S1-AVF per time step is uniform in ε ∈ (0, 1]. To obtain an explicit scheme, we consider the following approximation.…”

“…Among them, the Boris method [3] proposed in 1970 is still widely used by physicists, followed by some recent numerical analysis work [23,40] to address its mathematical property. Later on, many other structuralpreserving schemes have been designed, including the volume-preserving algorithm [28], the timesymmetric algorithm [24], the symplectic or K-symplectic algorithms [29,39,43,45,46], the Poisson integrators [30] and the energy-preserving algorithms [4,33,34].…”

“…More importantly, in the scheme the stiffness is not involved in the nonlinear equation thanks to splitting, and so the nonlinear solver can perform efficiently for all ε ∈ (0, 1]. In contrast, the other energy-preserving schemes such as the direct average vector field method [36], energy-preserving collocation methods [20], energyconserving line integral methods [4] and those from [33,34] quickly lose efficiency as ε decreases because of the stiffness in the nonlinear equation. On the other hand, under the maximal ordering scaling case of (1.1), we shall for the first time establish the rigorous optimal convergence result for a class of Lie-Trotter type splitting schemes including the proposed energy-preserving splitting and a volume-preserving splitting from the literature [11].…”

An explicit and practically invariants-preserving method for conservative systems

“…where the corresponding energy becomes H(x, v) = mv • v/2 + qϕ(x). Though several structure-preserving methods have been proposed for numerical solving (3.6), including symplectic methods [60,53], volumepreserving methods [46,34,26] and energy-conserving methods [41,9], with respect to the structures (…”

On Symmetrical Methods for Charged Particle Dynamics

“…In this paper, we are interested in investigating high-order numerical methods for solving the following charged particle dynamics [1][2][3][4]: q = p,ṗ = p × L(q) − ∇U(q), q(0) = q 0 , p(0) = p 0 , (q 0 , p 0 ) ∈ Ω ⊆ R 3 × R 3 , (1) where L(q) and −∇U(q) are, respectively, the magnetic and electric fields. In recent years, many energy-conserving numerical methods have been investigated within the framework of line integral methods [1,5] or Runge-Kutta methods [6][7][8][9].…”