Projection-Free Structure-Preserving Explicit Integrators on Manifolds
Wai Ming Chau · 2025
We introduce two new explicit numerical integrators: Manifold-TVDRK (MTV-DRK) and Manifold Taylor Series (MTS) methods, specifically designed for solving differential equations on manifolds, particularly on Sn and matrix Lie groups such as the special orthogonal group SO(n). Unlike many existing methods, our integrators do not require projection onto the manifold and do not involve local parameterization, making them both efficient and straightforward to implement. These integrators preserve geometric structures, such as orthonormality, and demonstrate excellent stability and accuracy. We provide comprehensive convergence analyses and benchmark our methods against classical Lie group integrators, highlighting their superior performance. Furthermore, we apply our methods to compute the ray tracing solution of the Eikonal equation on spheres and develop an adaptive framework to maintain the resolution of the wavefront. Another application involves computing Lyapunov exponents in dynamical systems. Both applications illustrate the practical robustness and wide applicability of our methods.