A Spherical Crank-Nicolson Integrator Based on the Exponential Map and the Spherical Linear Interpolation
Shingyu Leung
TL;DR
Addressing stiff ODEs on the unit sphere $\mathbb{S}^2$, the paper develops implicit, geometry-respecting integrators that avoid projection. By blending the exponential map with SLERP, it constructs three schemes—Spherical Backward Euler (SBE), projected BE (PBE), and a second-order spherical Crank-Nicolson (SCN)—each solved via Newton iterations. The SCN method is shown to be time-reversible and second-order accurate, with demonstrated energy preservation in Hamiltonian-like spherical flows, outperforming first-order baselines in convergence and stability tests. These methods provide robust, constraint-preserving tools for sphere-constrained dynamics with potential applications in rigid-body and geometric-ODE simulations.
Abstract
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.