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Splitting methods for unbounded operators

Arieh Iserles, Karolina Kropielnicka

TL;DR

This paper considers computational methods that split a vector field into three components in the case when both the vector field and the split components might be unbounded and derives an error expression and an error bound which are however valid in the presence of unbounded operators.

Abstract

This paper considers computational methods that split a vector field into three components in the case when both the vector field and the split components might be unbounded. We first employ classical Taylor expansion which, after some algebra, results in an expression for a second-order splitting which, strictly speaking, makes sense only for bounded operators. Next, using an alternative approach, we derive an error expression and an error bound in the same setting which are however valid in the presence of unbounded operators. While the paper itself is concerned with second-order splittings using three components, the method of proof in the presence of unboundedness remains valid (although significantly more complicated) in a more general scenario, which will be the subject of a forthcoming paper.

Splitting methods for unbounded operators

TL;DR

This paper considers computational methods that split a vector field into three components in the case when both the vector field and the split components might be unbounded and derives an error expression and an error bound which are however valid in the presence of unbounded operators.

Abstract

This paper considers computational methods that split a vector field into three components in the case when both the vector field and the split components might be unbounded. We first employ classical Taylor expansion which, after some algebra, results in an expression for a second-order splitting which, strictly speaking, makes sense only for bounded operators. Next, using an alternative approach, we derive an error expression and an error bound in the same setting which are however valid in the presence of unbounded operators. While the paper itself is concerned with second-order splittings using three components, the method of proof in the presence of unboundedness remains valid (although significantly more complicated) in a more general scenario, which will be the subject of a forthcoming paper.
Paper Structure (3 sections, 2 theorems, 49 equations)

This paper contains 3 sections, 2 theorems, 49 equations.

Table of Contents

  1. The problem
  2. Classical approach
  3. An iterated Duhamel approach

Key Result

Theorem 3.1

Subject to the consistency condition $P_1+P_2+P_3=\mathcal{L}$, the splitting $\hbox{\boldmath$S$\unboldmath}$ is of order three if and only if are bounded and the condition (eq:2.1), namely holds. In that case the error committed by the splitting is

Theorems & Definitions (2)

  • Theorem 3.1
  • Proposition 3.2