On alternating-conjugate splitting methods

TL;DR

The paper introduces alternating-conjugate splitting methods, formed by concatenating a complex-coefficient composition with its complex conjugate, to attain superior long-time behavior for linear unitary, Hermitian, and Hamiltonian flows. It provides a spectral and perturbation-theoretic framework showing AC schemes preserve essential invariants and remain close to unitary or symplectic dynamics, with rigorous results for simple-spectrum and Hermitian cases. New higher-order AC schemes up to order 6 are constructed, employing order-conditions and minimal-stage designs, and their efficiency is demonstrated through numerical experiments on linear Hamiltonian and unitary problems. The work suggests AC schemes as robust geometric integrators, with potential extensions to nonlinear problems where analyticity and structure-preserving properties can be leveraged, and highlights the trade-offs between real versus complex coefficients in practice.

Abstract

The new class of alternating-conjugate splitting methods is presented and analyzed. They are obtained by concatenating a given composition involving complex coefficients with the same composition but with the complex conjugate coefficients. We show that schemes of this type exhibit a good long time behavior when applied to linear unitary and linear Hamiltonian systems, in contrast with other methods based on complex coefficients, and study in detail their preservation properties. We also present new schemes within this class up to order 6 that exhibit a better efficiency than state-of-the-art splitting methods with real coefficients for some classes of problems.

Figures (6)

• Figure 1: Left panel: function $D_h$ defined by \ref{['dh']} vs. the step size $h$ obtained by schemes \ref{['scheme3']} and \ref{['schemes4']} applied to eq. \ref{['uf.1']} when $H$ is a $10 \times 10$real symmetric matrix with simple eigenvalues. $\Psi_{h}^{[3]}$ (black solid) and $\Psi_{h,sc}^{[4]}$ (blue dashed) are symmetric-conjugate, whereas $\Psi_{h,p}^{[4]}$ (red dash-dotted) is palindromic. Right panel: results achieved by the alternating-conjugate splitting methods \ref{['ac.1']} on the same problem. Now all methods behave as unitary maps for small enough $h$.
• Figure 2: Same as Figure \ref{['figure.1']} when $H$ is a $10 \times 10$complex Hermitian matrix with simple eigenvalues. The schemes depicted are $\Psi_{h}^{[3]}$ (black solid), $\Psi_{h,sc}^{[4]}$ (blue dashed) and $\Psi_{h,p}^{[4]}$ (red dash-dotted). Right panel corresponds to the alternating-conjugate version of the previous methods \ref{['ac.1']}. In this case, symmetric-conjugate methods do not provide a correct qualitative description of the system.
• Figure 3: Same as Figures \ref{['figure.1']} and \ref{['figure.2']} when $H$ is a $10 \times 10$complex Hermitian matrix with repeated eigenvalues. The schemes depicted are $\Psi_{h}^{[3]}$ (black solid), $\Psi_{h,sc}^{[4]}$ (blue dashed) and $\Psi_{h,p}^{[4]}$ (red dash-dotted). Right panel corresponds to the alternating-conjugate version of the previous methods \ref{['ac.1']}. Only alternating-conjugate methods behave as unitary integrators in this case.
• Figure 4: Imaginary part of the numerical solution as a function of time for a linear Hamiltonian system with $N=3$ and step size $h=2.5$. Left panel: schemes $\Psi_{h}^{[3]}$ (black solid), $\Psi_{h,sc}^{[4]}$ (blue dashed) and $\Psi_{h,p}^{[4]}$ (red dash-dotted). Right panel: results achieved by the alternating-conjugate version \ref{['ac.1']} on the same problem. Only the numerical solution obtained by alternating-conjugate methods remains almost real for very long times.
• Figure 5: Norm of the numerical solution (left) and error in energy (right) as a function of time for a linear Hamiltonian system with $N=3$ and step size $h=2.5$ for the following schemes: $\Psi_{h}^{[3]}$ (black solid), $\Psi_{h,sc}^{[4]}$ (blue dashed), $\Psi_{h,p}^{[4]}$ (red dash-dotted) and the new 4th-order alternating-conjugate method \ref{['nac4']} (green dashed). Only alternating-conjugate schemes provide a correct description of the dynamics.

Theorems & Definitions (21)

• Lemma 3.1
• proof
• Theorem 3.2
• Corollary 3.3
• proof : Proof of Theorem \ref{['thm:unitary']}
• Theorem 3.4
• proof
• Theorem 4.1
• Corollary 4.2
• proof : Proof of Corollary \ref{['cor:nice']}