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Energy-preserving iteration schemes for Gauss collocation integrators

Stefan Maier, Nicole Marheineke, Andreas Frommer

TL;DR

This paper presents energy-preserving iteration schemes for Gauss collocation integrators applied to Poisson systems with quadratic Hamiltonians. It establishes a $Q$-Arnoldi Krylov subspace method that preserves the $Q$-norm energy at every iterate, enabling early termination without sacrificing the invariant, and compares it to GMRES for linear problems. For nonlinear Poisson systems, it leverages Cayley-transform-based fixed-point and Newton-like solvers for the implicit midpoint rule, ensuring energy preservation along all iterates and achieving competitive convergence orders. The results show that the proposed linear and nonlinear energy-preserving solvers enable efficient, stable time integration for high-dimensional problems, maintaining the invariant with machine-precision accuracy across steps and time horizons.

Abstract

In this work, we develop energy-preserving iterative schemes for the (non-)linear systems arising in the Gauss integration of Poisson systems with quadratic Hamiltonian. Exploiting the relation between Gauss collocation integrators and diagonal Padé approximations, we establish a Krylov-subspace iteration scheme based on a $Q$-Arnoldi process for linear systems that provides energy conservation not only at convergence --as standard iteration schemes do--, but also at the level of the individual iterates. It is competitive with GMRES in terms of accuracy and cost for a single iteration step and hence offers significant efficiency gains, when it comes to time integration of high-dimensional Poisson systems within given error tolerances. On top of the linear results, we consider non-linear Poisson systems and design non-linear solvers for the implicit midpoint rule (Gauss integrator of second order), using the fact that the associated Padé approximation is a Cayley transformation. For the non-linear systems arising at each time step, we propose fixed-point and Newton-type iteration schemes that inherit the convergence order with comparable cost from their classical versions, but have energy-preserving iterates.

Energy-preserving iteration schemes for Gauss collocation integrators

TL;DR

This paper presents energy-preserving iteration schemes for Gauss collocation integrators applied to Poisson systems with quadratic Hamiltonians. It establishes a -Arnoldi Krylov subspace method that preserves the -norm energy at every iterate, enabling early termination without sacrificing the invariant, and compares it to GMRES for linear problems. For nonlinear Poisson systems, it leverages Cayley-transform-based fixed-point and Newton-like solvers for the implicit midpoint rule, ensuring energy preservation along all iterates and achieving competitive convergence orders. The results show that the proposed linear and nonlinear energy-preserving solvers enable efficient, stable time integration for high-dimensional problems, maintaining the invariant with machine-precision accuracy across steps and time horizons.

Abstract

In this work, we develop energy-preserving iterative schemes for the (non-)linear systems arising in the Gauss integration of Poisson systems with quadratic Hamiltonian. Exploiting the relation between Gauss collocation integrators and diagonal Padé approximations, we establish a Krylov-subspace iteration scheme based on a -Arnoldi process for linear systems that provides energy conservation not only at convergence --as standard iteration schemes do--, but also at the level of the individual iterates. It is competitive with GMRES in terms of accuracy and cost for a single iteration step and hence offers significant efficiency gains, when it comes to time integration of high-dimensional Poisson systems within given error tolerances. On top of the linear results, we consider non-linear Poisson systems and design non-linear solvers for the implicit midpoint rule (Gauss integrator of second order), using the fact that the associated Padé approximation is a Cayley transformation. For the non-linear systems arising at each time step, we propose fixed-point and Newton-type iteration schemes that inherit the convergence order with comparable cost from their classical versions, but have energy-preserving iterates.

Paper Structure

This paper contains 16 sections, 8 theorems, 53 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Padé approximations
  3. Krylov-subspace iteration schemes
  4. Krylov subspaces and the Arnoldi approximation
  5. Gauss integrators up to order six
  6. Arnoldi approximation of the exponential
  7. Numerical results for a linear Poisson system
  8. Convergence and energy preservation
  9. Long-term performance
  10. Cayley Transformations in Non-linear Solvers
  11. Fixed-point iteration
  12. Newton-like methods
  13. Numerical results for a non-linear system
  14. Convergence and energy preservation
  15. Long-term behavior
  16. ...and 1 more sections

Key Result

Proposition 1

All poles $\tau_j$ of $\mathcal{R}_s(z)$, i.e. the zeros of $\mathcal{D}_s(-z)$, are simple and located in the open right half plane, i.e. $\Re(\tau_j) > 0$. Moreover, in the partial fractional decomposition of $\mathcal{R}_s(z)$, the poles $\tau_j$ and coefficients $\omega_j$ appear in complex conjugate pairs or are real-valued.

Figures (4)

  • Figure 4.1: Performance of $Q$-Arnoldi approximation (QAA) and GMRES in the computation of $y_1$ for the mass-spring chain model. Left: Accuracy of iterates $x_k$ w.r.t. Euclidian norm of the residual $r_k = \mathcal{D}_{s}(-hJQ)x_k - \mathcal{D}_{s}(hJQ)y_0$. Right: Deviation from energy-conservation, i.e. $| 1 - \lVert x_k \rVert_Q/\lVert y_0 \rVert_Q|$. Top to bottom: Gauss integrators with $s=1$, $s=2$ and $s=3$; $h = 0.1$.
  • Figure 4.2: Performance of QAA and GMRES in Gauss integration of the mass-spring chain model over time interval $[0,1]$ for different step sizes $h$. Left: Convergence behavior of integrators w.r.t. $L^2([0,1])$-error. Middle: Absolute deviation from energy-conservation, i.e. $\max_i| 1 - \lVert y_i \rVert_Q/\lVert y_0 \rVert_Q|$. Right: Averaged number of Krylov subspace iterations per time step. Top to bottom: Gauss integrators with $s=1$, $s=2$ and $s=3$.
  • Figure 6.3: Performance of fixed point iteration (FP) and Cayley-BFGS method in the computation of $y_1$ for the rigid body model; $h=0.1$. Left: Accuracy of iterates w.r.t. Euclidean norm of the residual $\widetilde{r}$ from \ref{['eq:nonlinear_residual']}. Right: Deviation from energy-conservation.
  • Figure 6.4: Performance of FP and Cayley-BFGS in Gauss integration (midpoint rule) of the rigid body model over time interval $[0,1]$ for different step sizes $h$. Left: Convergence behavior of midpoint rule w.r.t. $L^2([0,1])$-error. Middle: Absolute deviation from energy-conservation, i.e. $\max_i| 1 - \lVert y_i \rVert_Q/\lVert y_0 \rVert_Q|$. Right: Averaged number of outer iterations per time step.

Theorems & Definitions (15)

  • Proposition 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • proof
  • Theorem 6
  • ...and 5 more