A Systematic Review of Spectral methods for numerical simulation of high-energy particle interactions: Methods, Architectures, and Future Research Directions

Abstract

The numerical simulation of high-energy particle interactions constitutes a fundamental challenge in computational physics, particularly in domains such as quantum chromodynamics, plasma physics, and particle accelerator modeling. Spectral methods have emerged as a powerful class of numerical techniques capable of delivering high accuracy and exponential convergence for smooth solutions, making them particularly suitable for resolving complex wave–particle dynamics and nonlinear interactions at high energies. This paper presents a systematic review of spectral methods applied to high-energy particle simulations, focusing on methodological developments, computational architectures, and emerging research trends. The review synthesizes recent advancements from 2018 to 2025, including Fourier spectral methods, Chebyshev-based approaches, discontinuous spectral element formulations, and hybrid AI-integrated frameworks. Key findings indicate that while spectral methods offer superior precision and reduced numerical dispersion, they face challenges related to scalability, handling discontinuities, and integration with heterogeneous computing environments. The paper contributes a structured analysis of existing literature, identifies critical research gaps, and proposes future directions emphasizing AI-assisted solvers, adaptive spectral frameworks, and secure computational pipelines.