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High-order Gauss-Legendre methods admit a composition representation and a conjugate-symplectic counterpart

Felice Iavernaro, Francesca Mazzia, Ernst Hairer

TL;DR

The paper addresses whether higher-order Gauss–Legendre Runge–Kutta methods admit a composition representation using multiderivative Euler-type steps and identifies a conjugate-symplectic partner. It first analyzes the two-stage, fourth-order Gauss method via discrete approximations of $D_1 f$ and $D_2 f$, producing two 2-stage RK methods whose half-step composition yields the Gauss method of order four and whose reverse composition yields a conjugate-symplectic partner with the same stability profile for the Dahlquist test equation. With a collocation-based construction at a degree-2, the MDMP4/MDTR4 variants reproduce the Gauss method and its conjugate-symplectic twin, and the framework generalizes to arbitrary order via a structured composition of derived RK methods. The results preserve geometric structure, require only a single nonlinear solve per step, and provide a fourth-order dense output, suggesting a practical pathway to high-order, structure-preserving integrators applicable to Hamiltonian systems.

Abstract

One of the most classical pairs of symplectic and conjugate-symplectic schemes is given by the Midpoint method (the Gauss-Runge-Kutta method of order 2) and the Trapezoidal rule. These can be interpreted as compositions of the Implicit and Explicit Euler methods, taken in direct and reverse order, respectively. This naturally raises the question of whether a similar composition structure exists for higher-order Gauss-Legendre methods. In this paper, we provide a positive answer by first examining the fourth-order case and then outlining a generalization to higher orders.

High-order Gauss-Legendre methods admit a composition representation and a conjugate-symplectic counterpart

TL;DR

The paper addresses whether higher-order Gauss–Legendre Runge–Kutta methods admit a composition representation using multiderivative Euler-type steps and identifies a conjugate-symplectic partner. It first analyzes the two-stage, fourth-order Gauss method via discrete approximations of and , producing two 2-stage RK methods whose half-step composition yields the Gauss method of order four and whose reverse composition yields a conjugate-symplectic partner with the same stability profile for the Dahlquist test equation. With a collocation-based construction at a degree-2, the MDMP4/MDTR4 variants reproduce the Gauss method and its conjugate-symplectic twin, and the framework generalizes to arbitrary order via a structured composition of derived RK methods. The results preserve geometric structure, require only a single nonlinear solve per step, and provide a fourth-order dense output, suggesting a practical pathway to high-order, structure-preserving integrators applicable to Hamiltonian systems.

Abstract

One of the most classical pairs of symplectic and conjugate-symplectic schemes is given by the Midpoint method (the Gauss-Runge-Kutta method of order 2) and the Trapezoidal rule. These can be interpreted as compositions of the Implicit and Explicit Euler methods, taken in direct and reverse order, respectively. This naturally raises the question of whether a similar composition structure exists for higher-order Gauss-Legendre methods. In this paper, we provide a positive answer by first examining the fourth-order case and then outlining a generalization to higher orders.

Paper Structure

This paper contains 6 sections, 3 theorems, 20 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and main result
  2. Background
  3. An MDMP4 variant based on a quadratic collocation polynomial
  4. An MDTR4 variant based on a quadratic collocation polynomial
  5. Generalization to higher-order formulae
  6. Conclusions

Key Result

Theorem 1

The Runge--Kutta scheme defined by the tableau MIDPOINT_COLL is symplectic if and only if $\alpha = \sqrt{3}/6$. In this case, the method coincides with the Gauss--Legendre method of order 4 and can be obtained as the composition $\Phi_{h/2} \circ \Psi_{h/2}$, where $\Phi_h$ and $\Psi_h$ are defined

Figures (1)

  • Figure 1: Maximum absolute error in the continuous extensions for the polynomial in (\ref{['dense']}) (solid blue line) and the collocation polynomial (dotted red line)).

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • proof