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An Explicit Symmetric Exponential Integrator and Its Error Estimate for the Relativistic Charged-Particle Dynamics

Zhirui Shen, Bin Wang

Abstract

This paper investigates the equations of motion for a relativistic charged particle in a general magnetic field. By reformulating the dynamics in four-dimensional spacetime and separating the linear and nonlinear parts, we construct an explicit symmetric exponential integrator based on Lie splitting. Rigorous analysis establishes its unconditional stability and second-order convergence. Numerical experiments confirm its superior performance, including accuracy, effciency and long-time Hamiltonian conservation.

An Explicit Symmetric Exponential Integrator and Its Error Estimate for the Relativistic Charged-Particle Dynamics

Abstract

This paper investigates the equations of motion for a relativistic charged particle in a general magnetic field. By reformulating the dynamics in four-dimensional spacetime and separating the linear and nonlinear parts, we construct an explicit symmetric exponential integrator based on Lie splitting. Rigorous analysis establishes its unconditional stability and second-order convergence. Numerical experiments confirm its superior performance, including accuracy, effciency and long-time Hamiltonian conservation.
Paper Structure (6 sections, 2 theorems, 23 equations, 3 figures)

This paper contains 6 sections, 2 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. A brief introduction to the 4D equations and sEI
  4. Convergence analysis for sEI
  5. Numerical Results
  6. Conclusion

Key Result

Proposition 2.1

Suppose $V_1,V_2,W_1,W_2 \in C^1(\mathbb{R}^4\times\mathbb{R}^4)$ and $h\in(0,1]$, and denote $\mathcal{V}=(V_1,V_2)^T$ and $\mathcal{W}=(W_1,W_2)^T$. Then the sEI (sEI) satisfies, with $C>0$ independent to $h$,

Figures (3)

  • Figure 1: Global errors $\mathrm{err}_U(1)$ under initial conditions (I) and (II).
  • Figure 2: The comparison of the CPU time between sEI and Heun under initial conditions (I) and (II).
  • Figure 3: The relative errors of Hamiltonian for sEI and Heun method with $h=2^{-6}$.

Theorems & Definitions (4)

  • Proposition 2.1: Stability
  • proof
  • Theorem 2.2
  • proof