[Paper Review] Accurate numerical simulation of radiation reaction effects in strong electromagnetic fields

N. Elkina, A. M. Fedotov+2 more|arXiv (Cornell University)|Jan 30, 2014

Superconducting Materials and Applications5 references3 citations

TL;DR

This paper presents a high-accuracy numerical solver for radiation reaction in strong electromagnetic fields using implicit Runge-Kutta-Nyström (RKN) methods applied to the covariant Landau-Lifshitz equation. It demonstrates that implicit collocation RKN methods of order four, six, and eight outperform explicit methods in conserving the mass-shell condition and energy, enabling precise long-term simulation of electron dynamics in intense laser fields, with applications showing reduced particle penetration into focal regions due to radiation reaction.

ABSTRACT

The Landau-Lifshitz equation provides an efficient way to account for the effects of radiation reaction without acquiring the non-physical solutions typical for the Lorentz-Abraham-Dirac equation. We solve the Landau-Lifshitz equation in its covariant four-vector form in order to control both the energy and momentum of radiating particle. Our study reveals that implicit time-symmetric collocation methods of the Runge-Kutta-Nyström type are superior in both accuracy and better maintaining the mass-shell condition than their explicit counterparts. We carry out an extensive study of numerical accuracy by comparing the analytical and numerical solutions of the Landau-Lifshitz equation. Finally, we present the results of simulation of particles scattering by a focused laser pulse. Due to radiation reaction, particles are less capable for penetration into the focal region, as compared to the case of radiation reaction neglected. Our results are important for designing the forthcoming experiments with high intensity laser fields.

Motivation & Objective

• To develop a numerically stable and accurate method for simulating radiation reaction in strong electromagnetic fields.
• To overcome the limitations of the Lorentz-Abraham-Dirac equation, such as runaway solutions and causality violations.
• To improve upon explicit integrators by employing implicit, time-symmetric, collocation-based RKN methods that conserve quadratic invariants.
• To validate the method against analytical solutions in constant magnetic fields and plane waves.
• To apply the solver to realistic scenarios, such as electron scattering in focused laser pulses.

Proposed method

• The Landau-Lifshitz equation is formulated in covariant four-vector form to ensure consistent energy and momentum conservation.
• Implicit, collocation-based Gauss-Legendre Runge-Kutta-Nyström (RKN) methods of order four, six, and eight are derived using Gaussian quadrature nodes.
• The methods are constructed via collocation points to ensure time symmetry and exact conservation of quadratic invariants like the Minkowski norm.
• The numerical scheme is validated against analytical solutions for particle motion in a constant magnetic field and in a circularly polarized plane wave.
• An adaptive time step control is suggested to mitigate accuracy loss in regions of strong field gradients.
• The method is applied to simulate relativistic electron scattering in focused laser beams, comparing results with and without radiation reaction.

Experimental results

Research questions

• RQ1 Can implicit, time-symmetric RKN methods provide superior accuracy and conservation properties for simulating radiation reaction in strong fields?
• RQ2 How do implicit RKN methods compare to explicit methods in preserving the mass-shell condition and energy conservation?
• RQ3 What is the impact of radiation reaction on electron scattering dynamics in focused laser pulses?
• RQ4 How does numerical accuracy correlate with the impact parameter in high-gradient field regions?
• RQ5 Can the Landau-Lifshitz equation with implicit RKN integration accurately model radiation reaction effects comparable to quasiclassical Monte Carlo simulations?

Key findings

• Implicit collocation RKN methods of order four, six, and eight demonstrate significantly higher accuracy than explicit methods in solving the Landau-Lifshull equation.
• These implicit methods better preserve the mass-shell condition $u_{ u}u^{ u} = c^2$ over long integration times.
• In simulations of electron scattering by a focused laser pulse, radiation reaction reduces particle penetration into the focal region compared to the non-radiating case.
• The numerical accuracy degrades near strong field gradients, particularly for larger impact parameters, indicating the need for adaptive time stepping.
• The results from the Landau-Lifshitz solver show qualitative agreement with quasiclassical Monte Carlo simulations of hard photon emission.
• The use of higher-order implicit RKN methods allows larger time steps while maintaining high accuracy, improving computational efficiency despite higher per-step cost.