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Fourth-order compact exponential splittings for unbounded operators

Juan Carlos Del Valle, Arieh Iserles, Karolina Kropielnicka

TL;DR

The paper develops and rigorously analyzes a fourth-order, compact exponential splitting $\mathcal{T}_{ACB}^{(4)}$ for evolution problems with an unbounded operator $A$ and a bounded, time-dependent $B(t)$, relevant for quantum mechanics and Schrödinger-type equations. It derives the method via iterated Duhamel formula and Hermite–Birkhoff quadratures, reconstructs the nontrivial semigroup term involving $[B(t),[A,B(t)]]$, and proves fourth-order convergence under precise commutator and differentiability assumptions, while revealing that no single $\tau$ minimizes all error components. The analysis renders explicit error bounds for truncation and quadrature errors and shows that the optimal $\tau$ closely matches the cubic Gauss–Legendre point $\tau_{opt}=\tfrac{1}{2}-\tfrac{\sqrt{15}}{10}$. A numerical Schrödinger example corroborates the theoretical predictions, demonstrating substantial accuracy gains with the optimal parameter and providing a practical guideline for parameter choice in unbounded-domain simulations.

Abstract

We present a derivation and error bound for the family of fourth order splittings, originally introduced by Chin and Chen, where one of the operators is unbounded and the second one bounded but time dependent, and which are dependent on a parameter. We first express the error by an iterated application of the Duhamel principle, followed by quadratures of Birkhoff-Hermite type of the underlying multivariate integrals. This leads to error estimates and bounds, derived using Peano/Sard kernels and direct estimates of the leading error term. Our analysis demonstrates that, although no single value of the parameter can minimise simultaneously all error components, an excellent compromise is the cubic Gauss--Legendre point $1/2-\sqrt{15}/10$.

Fourth-order compact exponential splittings for unbounded operators

TL;DR

The paper develops and rigorously analyzes a fourth-order, compact exponential splitting for evolution problems with an unbounded operator and a bounded, time-dependent , relevant for quantum mechanics and Schrödinger-type equations. It derives the method via iterated Duhamel formula and Hermite–Birkhoff quadratures, reconstructs the nontrivial semigroup term involving , and proves fourth-order convergence under precise commutator and differentiability assumptions, while revealing that no single minimizes all error components. The analysis renders explicit error bounds for truncation and quadrature errors and shows that the optimal closely matches the cubic Gauss–Legendre point . A numerical Schrödinger example corroborates the theoretical predictions, demonstrating substantial accuracy gains with the optimal parameter and providing a practical guideline for parameter choice in unbounded-domain simulations.

Abstract

We present a derivation and error bound for the family of fourth order splittings, originally introduced by Chin and Chen, where one of the operators is unbounded and the second one bounded but time dependent, and which are dependent on a parameter. We first express the error by an iterated application of the Duhamel principle, followed by quadratures of Birkhoff-Hermite type of the underlying multivariate integrals. This leads to error estimates and bounds, derived using Peano/Sard kernels and direct estimates of the leading error term. Our analysis demonstrates that, although no single value of the parameter can minimise simultaneously all error components, an excellent compromise is the cubic Gauss--Legendre point .

Paper Structure

This paper contains 14 sections, 2 theorems, 70 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The derivation of the method
  3. A general pattern of quadrature
  4. Error Analysis
  5. The first two error bounds
  6. Quadrature errors
  7. The 1D Integral
  8. The 2D Integral
  9. The 3D and 4D Integrals
  10. A numerical example
  11. Conclusions
  12. Acknowledgements
  13. Explicit quadrature formulæ
  14. Explicit terms of the type (\ref{['NiceTerm']})

Key Result

Lemma 3.1

For sufficiently smooth $B(t)$ the $m$th derivative of $\mathrm{e}^{-tA}B(t)\mathrm{e}^{tA}$ with respect to $t$ can be expressed in the form This implies that for $n=1,\ldots,4$

Figures (3)

  • Figure 1: Two- and three-dimensional simplexes displaying the nodes for $\tau=0$.
  • Figure 2: The 1D Peano kernel (\ref{['kernel']}) for various values of $\tau$: $\tau=k/16$ for $k=1,\ldots,7$ in increasingly darker shades of blue and $\tau=(5-\sqrt{15})/10$ as a thick red line.
  • Figure 3: Global Error (in log-log scale) for representative values of $\tau$ as a function of the time step $h$. The black-dashed line represents the plot of $h^4$. The values $\tau=0.1127$ and $0.1$ results in similar global errors.

Theorems & Definitions (2)

  • Lemma 3.1
  • Theorem 3.3