Goodbye Christoffel Symbols: A Flexible and Efficient Approach for Solving Physical Problems in Curved Spaces

Abstract

Traditional methods for solving physical equations in curved spaces, especially in fluid mechanics and general relativity, rely heavily on the use of Christoffel symbols. These symbols provide the necessary corrections to account for curvature in differential geometries but lead to significant computational complexity, particularly in numerical simulations. In this paper, we propose a novel, simplified approach that obviates the need for Christoffel symbols by symbolic programming and advanced numerical methods. Our approach is based on defining a symbolic mapping between Euclidean space and curved coordinate systems, enabling the transformation of spatial and temporal derivatives through Jacobians and their inverses. This eliminates the necessity of using Christoffel symbols for defining local bases and tensors, allowing for the direct application of physical laws in Cartesian coordinates even when solving problems in curved spaces. We demonstrate the robustness and flexibility of our method through several examples, including the derivation of the Navier-Stokes equations in cylindrical coordinates, the modeling of complex flows in bent cylindrical tubes, and the breakup of viscoelastic fluid threads. These examples highlight how our method simplifies the numerical formulation while maintaining accuracy and efficiency. Additionally, we explore how these advancements benefit free-surface flows, where mapping physical 3D domains to a simpler computational domain is essential for solving moving boundary problems.