Generalized extrapolation methods based on compositions of a basic 2nd-order scheme

TL;DR

The paper develops a generalized framework for high-order integrators built from linear combinations of compositions of a time-symmetric 2nd-order map $S_h$, extending extrapolation and multi-product expansions. By employing Lie-operator formalism and Baker–Campbell–Hausdorff expansions, it derives order conditions and analytic coefficients for constructing order-4, -6, and -8 schemes, and introduces latency-aware variants that reduce communication in parallel implementations while preserving qualitative structure (e.g., pseudo-symplectic behavior) to higher orders. Numerical experiments on Kepler and Lotka–Volterra problems demonstrate variable performance gains depending on the problem and the amount of latency, with several optimized schemes delivering improved long-time behavior and conservation properties. The work also analyzes rounding errors and provides practical guidance for implementing increments-based summation to mitigate numerical noise, showing the methods’ potential for efficient, high-order, latency-aware integration of conservative dynamical systems.

Abstract

We propose new linear combinations of compositions of a basic second-order scheme with appropriately chosen coefficients to construct higher order numerical integrators for differential equations. They can be considered as a generalization of extrapolation methods and multi-product expansions. A general analysis is provided and new methods up to order 8 are built and tested. The new approach is shown to reduce the latency problem when implemented in a parallel environment and leads to schemes that are significantly more efficient than standard extrapolation when the linear combination is delayed by a number of steps.

Figures (6)

• Figure 1: Order 4. Efficiency diagrams (a), (b), and relative error in energy vs. time (c), (d) for the Kepler problem and the Lotka--Volterra system. All methods have $m=2$, so that they involve 2-stage compositions. In all four plots B4 and $\psi_h^{[3,2]}$, which (numerically) vanish the same terms in the error expansion, are virtually indistinguishable.
• Figure 2: Order 6. Efficiency diagrams (a), (b), and relative error in energy vs. time (c), (d) for the Kepler problem and the Lotka--Volterra system. All methods have $m=3$, and they involve 3-stage symmetric compositions. In all four plots B6 and the $k=5$ integrator, which both vanish $G_{71},\tilde{G}_{87},G_{91}$, are virtually indistinguishable.
• Figure 3: Order 6. Efficiency diagrams for both problems. The proposed scheme based on a linear combination of $k=4$ non-palindromic compositions shows an improved behavior on the Lotka--Volterra system.
• Figure 4: Order 8. Efficiency diagrams for both problems. Including an additional stage in the composition renders a better performance with respect to the standard MPE only for the Kepler problem.
• Figure 5: Order 4 when the sum is computed after $p$ steps. Extrapolation method and scheme $\psi_{h,s}^{[3,2]}$. The results obtained by the pseudo-symplectic method of order 7 do not depend on the value of $p$, in accordance with the analysis. The advantages become more apparent on increasing the final integration time $t_f$ (right panel).