2025/03/22 by Shingyu Leung, Leung, Shingyu
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2503.17618
openalex publication_date 2025/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose implicit integrators for solving stiff differential equations on unit spheres. Our approach extends the standard backward Euler and Crank-Nicolson methods in Cartesian space by incorporating the geometric constraint inherent to the unit sphere without additional projection steps to enforce the unit length constraint on the solution. We construct these algorithms using the exponential map and spherical linear interpolation (SLERP) formula on the unit sphere. Specifically, we introduce a spherical backward Euler method, a projected backward Euler method, and a second-order symplectic spherical Crank-Nicolson method. While all methods require solving a system of nonlinear equations to advance the solution to the next time step, these nonlinear systems can be efficiently solved using Newton's iterations. We will present several numerical examples to demonstrate the effectiveness and convergence of these numerical schemes. These examples will illustrate the advantages of our proposed methods in accurately capturing the dynamics of stiff systems on unit spheres.