Conditional Stability of the Euler Method on Riemannian Manifolds

We derive nonlinear stability results for numerical integrators on Riemannian manifolds, by imposing conditions on the ODE vector field and the step size that makes the numerical solution non-expansive whenever the exact solution is non-expansive over the same time step. Our model case is a geodesic version of the explicit Euler method. Precise bounds are obtained in the case of Riemannian manifolds of constant sectional curvature. The approach is based on a cocoercivity property of the vector field adapted to manifolds from Euclidean space. It allows us to compare the new results to the corresponding well-known results in flat spaces, and in general we find that a non-zero curvature will deteriorate the stability region of the geodesic Euler method. The step size bounds depend on the distance traveled over a step from the initial point. Numerical examples for spheres and hyperbolic 2-space confirm that the bounds are tight.

💡 Research Summary

This paper establishes a framework for analyzing the conditional stability of numerical integrators on Riemannian manifolds, focusing on a geodesic version of the explicit Euler method (GEE). The core objective is to derive conditions on the ODE vector field and the step size that guarantee the numerical solution is non-expansive (i.e., does not increase the distance between two solutions) whenever the exact flow has the same property over a single time step.

The authors adapt the concept of cocoercivity from Euclidean numerical analysis to the Riemannian setting. A vector field X is defined as α-cocoercive if its covariant derivative ∇X satisfies ⟨∇_v X, v⟩ ≤ -α∥∇_v X∥² for all tangent vectors v. This condition, weaker than monotonicity, is crucial for analyzing explicit schemes. They prove that if X is α-cocoercive on a geodesically convex set, then the exact flow is locally non-expansive.

The central analysis is performed for manifolds of constant sectional curvature ρ. The GEE step is interpreted as the endpoint of a geodesic variation. The difference between two such steps starting from two nearby points is governed by a Jacobi field J(t) along the geodesic connecting the initial points. For constant curvature, the Jacobi equation admits closed-form solutions involving trigonometric (ρ>0), linear (ρ=0), or hyperbolic (ρ<0) functions. By imposing that the norm of this Jacobi field does not increase from t=0 (initial tangent) to t=1 (end of the GEE step), the authors derive precise step-size restrictions for non-expansiveness.

The main results are two theorems. Theorem 6 handles positively curved manifolds (like spheres of radius r, where ρ=1/r²). It states that for the GEE map to be non-expansive, the step size h must satisfy h ≤ α * (κ / tan(κd)), where κ = ∥X_p∥√ρ and d is the initial distance between the two points. Theorem 9 provides the analogous condition for negatively curved (hyperbolic) spaces: h ≤ α * (κ / tanh(κd)). In both cases, the bound depends critically on the curvature ρ and the initial distance d. As the curvature magnitude increases (|ρ| larger), the permissible step size h shrinks, deteriorating the stability region compared to the flat Euclidean case (ρ=0), where the condition reduces to the classical h ≤ 2α. This quantitatively demonstrates the negative impact of curvature on the stability of explicit geometric integrators.

The paper includes numerical experiments on the sphere S², S³, and the hyperbolic plane H², confirming that the derived bounds are tight; exceeding them leads to expansive behavior of the numerical method. This work pioneers the application of cocoercivity-based stability analysis to Riemannian manifolds, providing a geometric tool to inform step-size selection for simulating dynamical systems on curved spaces, with potential applications in optimization, control, and computer vision.