Advanced Theoretical Analysis of Stability and Convergence in Computational Fluid Dynamics for Computer Graphics

TL;DR

This paper provides a rigorous theoretical analysis of stability, convergence, and physical property preservation in numerical fluid simulation methods for computer graphics, establishing foundational mathematical principles.

Contribution

It introduces new theoretical conditions for stability, convergence rates, and property preservation in fluid simulation schemes, advancing the mathematical understanding of these methods.

Findings

Derived stability conditions for semi-Lagrangian and particle methods.

Established convergence rates for Navier-Stokes discretizations.

Proved maintenance of incompressibility and vorticity conservation.

Abstract

Mathematical modeling of fluid dynamics for computer graphics requires high levels of theoretical rigor to ensure visually plausible and computationally efficient simulations. This paper presents an in-depth theoretical framework analyzing the mathematical properties, such as stability, convergence, and error bounds, of numerical schemes used in fluid simulation. Conditions for stability in semi-Lagrangian and particle-based methods were derived, demonstrating that these methods remain stable under certain conditions. Furthermore, convergence rates for Navier-Stokes discretizations were obtained, showing that numerical solutions converge to analytical solutions as spatial resolution and time step decrease. Furthermore, new theoretical results were introduced on the maintenance of incompressibility and conservation of vorticity, which are crucial for the physical accuracy of simulations. The findings serve as a mathematical foundation for future research in adaptive fluid simulation, guiding the development of robust simulation techniques for real-time graphics applications.