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Conditional Stability of the Euler Method on Riemannian Manifolds

Marta Ghirardelli, Brynjulf Owren, Elena Celledoni

TL;DR

This work addresses the stability of numerical integrators on Riemannian manifolds by developing an intrinsic cocoercivity framework and applying it to a geodesic variant of the explicit Euler method (GEE). It derives curvature-dependent step-size bounds that guarantee non-expansivity of the numerical flow whenever the exact flow is non-expansive, with precise results in spaces of constant sectional curvature. The main contributions include the first intrinsic stability analysis using only Riemannian distance, explicit bounds involving the Jacobi-field based functions $f_1$, $f_2$, $f_3$ and the parameter $\kappa = h \|X\| \sqrt{| ho|}$, and numerical validations on $S^2$, $S^3$, and $\mathbb{H}^2$ showing the bounds are tight. The results illuminate how curvature degrades stability regions for explicit geodesic methods, and open avenues for extending these techniques to other intrinsic integrators and more general manifolds.

Abstract

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.

Conditional Stability of the Euler Method on Riemannian Manifolds

TL;DR

This work addresses the stability of numerical integrators on Riemannian manifolds by developing an intrinsic cocoercivity framework and applying it to a geodesic variant of the explicit Euler method (GEE). It derives curvature-dependent step-size bounds that guarantee non-expansivity of the numerical flow whenever the exact flow is non-expansive, with precise results in spaces of constant sectional curvature. The main contributions include the first intrinsic stability analysis using only Riemannian distance, explicit bounds involving the Jacobi-field based functions , , and the parameter , and numerical validations on , , and showing the bounds are tight. The results illuminate how curvature degrades stability regions for explicit geodesic methods, and open avenues for extending these techniques to other intrinsic integrators and more general manifolds.

Abstract

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.

Paper Structure

This paper contains 18 sections, 7 theorems, 100 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Remarks on the history of stability in Euclidean space.
  3. Numerical integrators on manifolds.
  4. Related work.
  5. Our contributions.
  6. Background and preliminaries
  7. Conditional stability of the Geodesic Explicit Euler method in the constant curvature case.
  8. Examples
  9. An example on $S^2$
  10. Numerical results.
  11. An example on $\mathbb{H}^2$
  12. Numerical results.
  13. An example on $S^3$
  14. Numerical results.
  15. Conclusion and future work
  16. ...and 3 more sections

Key Result

Proposition 3

Let $(\mathcal{M}, g)$ be a Riemannian manifold, $\mathcal{U}\subset\mathcal{M}$ an open, geodesically convex set, and let $X\in\mathfrak{X}(\mathcal{M})$ be $\alpha$-cocoercive on $\mathcal{U}$ for some $\alpha > 0$. Then, for any $x_0, y_0\in\mathcal{U}$ there exists $t^*>0$ such that for $0<t\le t^*$.

Figures (6)

  • Figure 1: Visualization of the function appearing in Theorem \ref{['theo: GEE non-expansive for positive rho']}, $f_2(\kappa) - \sqrt{f_1(\kappa) f_3(\kappa)} = (1-\cos(\kappa)\,\mathrm{sinc\,}(\kappa)) - \sqrt{\sin^2(\kappa)(1-\,\mathrm{sinc\,}^2(\kappa))}$.
  • Figure 2: Construction for the proof of theorem \ref{['theo: GEE non-expansive for positive rho']}.
  • Figure 3: Visualization of the function appearing in Theorem \ref{['theo: GEE non-expansive for negative rho']}, $f_2(\kappa) - \sqrt{f_1(\kappa) f_3(\kappa)} = \left(\cosh(\kappa)\frac{\sinh(\kappa)}{\kappa}-1\right) - \sinh\kappa\sqrt{\frac{\sinh^2(\kappa)}{\kappa^2}-1}$. For large $\kappa$ it behaves as $\frac{\kappa}{2}-1$.
  • Figure 4: The plots show the largest stepsize $h$ found numerically versus the one given by condition \ref{['eq: condition on h for positive rho']} in Theorem \ref{['theo: GEE non-expansive for positive rho']}, for different initial elevation angles $\phi_0$ and different values of the parameter $\epsilon$.
  • Figure 5: The plots show the largest stepsize $h$ found numerically versus the estimated one given by solving numerically condition \ref{['eq: condition on h for negative rho']} in Theorem \ref{['theo: GEE non-expansive for negative rho']}, for different initial $y_0$ and different values of the parameter $\epsilon$.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Definition 1
  • Remark 2
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • ...and 9 more