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Error bounds for splitting methods in unitary problems

Fernando Casas, Ander Murua

Abstract

Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes are available that achieve high accuracy while preserving key qualitative properties of the underlying dynamical system, and are successfully used across a broad range of fields. In this work, we present a systematic analysis of both local and global errors arising from arbitrary splitting methods applied to unitary problems. Two complementary types of error estimates are derived. The first is expressed in terms of operator norms, while the second is formulated using norms of commutators and can, under suitable assumptions, be extended to certain classes of unbounded operators. Special attention is devoted to the case where only two operators are involved. The theoretical results are illustrated by deriving explicit error bounds for some representative schemes.

Error bounds for splitting methods in unitary problems

Abstract

Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes are available that achieve high accuracy while preserving key qualitative properties of the underlying dynamical system, and are successfully used across a broad range of fields. In this work, we present a systematic analysis of both local and global errors arising from arbitrary splitting methods applied to unitary problems. Two complementary types of error estimates are derived. The first is expressed in terms of operator norms, while the second is formulated using norms of commutators and can, under suitable assumptions, be extended to certain classes of unbounded operators. Special attention is devoted to the case where only two operators are involved. The theoretical results are illustrated by deriving explicit error bounds for some representative schemes.

Paper Structure

This paper contains 23 sections, 20 theorems, 203 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Error bounds depending on the norm of the operators
  3. General estimates of the local error
  4. Estimates for compositions of the Lie--Trotter and Strang schemes
  5. Estimates of the local error for symmetric schemes
  6. Global error
  7. Improved bounds based on the norm of the operators
  8. Series expansions with multi-indices
  9. Refined local error estimates for general sequences of coefficients
  10. Refined local error estimates for palindromic sequences of coefficients
  11. Numerical illustration of the refined estimates
  12. Error bounds in terms of commutators
  13. Estimates based on the modified equation satisfied by the splitting method
  14. (a) Lie--Trotter.
  15. (b) Strang splitting with two operators.
  16. ...and 8 more sections

Key Result

Proposition 2.1

If $C_1, \ldots, C_r$ are bounded skew-adjoint operators, then, for all $q \geq 1$, where $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Error bounds for the time-symmetric 6th-order scheme \ref{['sofro_spa']} with $s=13$. Lines correspond (from top to bottom) to the result provided by Corollary \ref{['th_compoS']} for a general composition of Strang splittings with $q=7$ (NS $q=7$), Corollary \ref{['th_compo_sym']} also with $q=7$ (S $q=7$), and the refined estimates provided by Corollary \ref{['th_compo_sym_refined']} for symmetric schemes with $q=9$ and $q=11$.

Theorems & Definitions (28)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Corollary 2.5
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • ...and 18 more