A Systematic Review of Hyperbolic PDE models for dynamo-type magnetic field behaviour: Methods, Architectures, and Future Research Directions

Abstract

Hyperbolic partial differential equation (PDE) models have emerged as a powerful mathematical framework for describing dynamo-type magnetic field behavior in complex physical systems, including astrophysical plasmas, geophysical flows, and engineered electromagnetic environments. Unlike parabolic formulations that emphasize diffusion-dominated processes, hyperbolic PDEs capture wave propagation, finite signal speeds, and transient dynamics that are essential for understanding magnetic field generation and evolution. This paper presents a comprehensive systematic review of hyperbolic PDE-based models for dynamo mechanisms, focusing on their mathematical formulations, computational architectures, and integration with modern computational paradigms such as machine learning and generative artificial intelligence. The study examines recent advances between 2018 and 2025, highlighting numerical schemes, stability considerations, and hybrid modeling approaches. Key findings indicate a growing shift toward high-resolution shock-capturing methods, physics-informed neural networks, and multi-scale coupling strategies that enhance predictive accuracy while maintaining computational efficiency. The review also identifies critical challenges, including stiffness handling, scalability, and uncertainty quantification. The primary contribution of this work lies in synthesizing interdisciplinary advancements, establishing connections between classical dynamo theory and emerging AI-driven methodologies, and outlining future research directions that emphasize robustness, real-time simulation, and integration into software engineering ecosystems.