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B-stability of numerical integrators on Riemannian manifolds

Martin Arnold, Elena Celledoni, Ergys Çokaj, Brynjulf Owren, Denise Tumiotto

Abstract

We propose a generalization of nonlinear stability of numerical one-step integrators to Riemannian manifolds in the spirit of Butcher's notion of B-stability. Taking inspiration from Simpson-Porco and Bullo, we introduce non-expansive systems on such manifolds and define B-stability of integrators. In this first exposition, we provide concrete results for a geodesic version of the Implicit Euler (GIE) scheme. We prove that the GIE method is B-stable on Riemannian manifolds with non-positive sectional curvature. We show through numerical examples that the GIE method is expansive when applied to a certain non-expansive vector field on the 2-sphere, and that the GIE method does not necessarily possess a unique solution for large enough step sizes. Finally, we derive a new improved global error estimate for general Lie group integrators.

B-stability of numerical integrators on Riemannian manifolds

Abstract

We propose a generalization of nonlinear stability of numerical one-step integrators to Riemannian manifolds in the spirit of Butcher's notion of B-stability. Taking inspiration from Simpson-Porco and Bullo, we introduce non-expansive systems on such manifolds and define B-stability of integrators. In this first exposition, we provide concrete results for a geodesic version of the Implicit Euler (GIE) scheme. We prove that the GIE method is B-stable on Riemannian manifolds with non-positive sectional curvature. We show through numerical examples that the GIE method is expansive when applied to a certain non-expansive vector field on the 2-sphere, and that the GIE method does not necessarily possess a unique solution for large enough step sizes. Finally, we derive a new improved global error estimate for general Lie group integrators.
Paper Structure (14 sections, 4 theorems, 44 equations, 6 figures)

This paper contains 14 sections, 4 theorems, 44 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Non-expansive systems
  3. Numerical integrators on manifolds and B-stability
  4. Geodesic Explicit Euler (GEE) method
  5. Geodesic Implicit Euler (GIE) method
  6. Geodesic Implicit Midpoint (GIMP) method
  7. The case with non-positive sectional curvature
  8. The case with positive sectional curvature: The 2-sphere
  9. Killing vector fields
  10. A Killing vector field on $S^2$
  11. A non-uniqueness issue
  12. Relation to other Lie group integrators
  13. A bound for the global error
  14. Conclusions and further work

Key Result

Theorem 2.2

Let $(M, g)$ be a Riemannian manifold, $\mathcal{U}\subset M$ a geodesically convex set, and let $X$ be a vector field on $M$ satisfying the monotonicity condition eq:X_contractive_in_U on $\mathcal{U}$ with a constant $\nu \in \mathbb{R}$. Suppose that for any $x_0, y_0 \in \mathcal{U}$, there is a

Figures (6)

  • Figure 1: Construction for the proofs of Theorems \ref{['thm:bound_exact_flow']} and \ref{['thm:b_stab_gie']}.
  • Figure 2: Riemannian distance of two solutions after one step plotted for increasing values of the step size $h$ with the same initial values.
  • Figure 3: One step of GIE method for two initial points with increasing step size $h$ (left) and their Riemannian distance (right).
  • Figure 4: Top: One step of SPHMP \ref{['eq:sphmp']} (left) and GIMP \ref{['eq:gimp']} (right) method for the same two initial points with increasing step size $h$. Bottom: Riemannian distance of two numerical solutions after one step plotted for increasing values of the step size $h$.
  • Figure 5: A bifurcation diagram for solutions to the equation \ref{['eq:y3eq']}.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • proof
  • Remark 2.4
  • Definition 2.5: Non-expansive system
  • Definition 2.6: B-stability
  • Theorem 3.1: B-stability of the GIE method
  • proof
  • Example 3.2
  • ...and 5 more