The recent developments in the research area have significantly advanced the simulation and analysis of complex physical phenomena, particularly in anisotropic diffusion processes and high-dimensional linear hyperbolic equations. Innovations in smoothed particle hydrodynamics (SPH) have enabled more accurate and robust solutions for anisotropic diffusion problems, addressing issues such as contaminant transport and fluid diffusion through porous membranes. These advancements leverage modified second derivative models and anisotropic kernel resolutions to achieve excellent agreement with theoretical solutions, demonstrating second-order accuracy and the suppression of spurious oscillations.

In parallel, there has been notable progress in the development of unified lattice Boltzmann models for high-dimensional linear hyperbolic equations. These models, based on the Bhatnagar-Gross-Krook (BGK) approach, have been refined to ensure fourth-order consistency and stability, with particular attention to boundary conditions and entropy stability. The theoretical convergence and stability analyses have been validated through numerical experiments, highlighting the superior stability of full-way boundary schemes over half-way schemes.

Noteworthy contributions include a novel SPH formulation that effectively handles anisotropic diffusion in complex scenarios, and a fourth-order BGK lattice Boltzmann model that enhances the accuracy and stability of high-dimensional linear hyperbolic equation solutions.