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Second-order exponential splittings in the presence of unbounded and time-dependent operators: construction and convergence

Karolina Kropielnicka, Juan Carlos del Valle

TL;DR

This paper addresses linear evolution equations with unbounded operators, $u'(t)=[A+B(t)]u(t)$, by reframing the problem through iterated Duhamel's formula and introducing two second-order exponential splitting families, $\mathcal{F}(h,\tau)$ and $\mathcal{D}(h,\tau)$. Quadrature rules, notably Birkhoff quadratures, are shown to be the building blocks that determine the splitting structure and the appearance of derivatives of $B(t)$ and commutators with $A$; error terms are derived and bounded, yielding conditions under which the methods achieve $\mathcal{O}(h^2)$ global accuracy or $\mathcal{O}(h^3)$ local accuracy. The paper provides explicit constructions, extensive error analysis, and numerical demonstrations on the Schrödinger and transport equations, illustrating the influence of $\tau$ on accuracy and cost. The framework both clarifies the link between splitting methods and exponential integrators for unbounded operators and points to feasible pathways for higher-order extensions using multidimensional Birkhoff quadratures.

Abstract

For linear differential equations of the form $u'(t)=[A + B(t)] u(t)$, $t\geq0$, with a possibly unbounded operator $A$, we construct and deduce error bounds for two families of second-order exponential splittings. The role of quadratures when integrating the twice-iterated Duhamel's formula is reformulated: we show that their choice defines the structure of the splitting. Furthermore, the reformulation allows us to consider quadratures based on the Birkhoff interpolation to obtain not only pure-stages splittings but also those containing derivatives of $B(t)$ and commutators of $A$ and $B(t)$. In this approach, the construction and error analysis of the splittings are carried out simultaneously. We discuss the accuracy of the members of the families. Numerical experiments are presented to complement the theoretical consideration.

Second-order exponential splittings in the presence of unbounded and time-dependent operators: construction and convergence

TL;DR

This paper addresses linear evolution equations with unbounded operators, , by reframing the problem through iterated Duhamel's formula and introducing two second-order exponential splitting families, and . Quadrature rules, notably Birkhoff quadratures, are shown to be the building blocks that determine the splitting structure and the appearance of derivatives of and commutators with ; error terms are derived and bounded, yielding conditions under which the methods achieve global accuracy or local accuracy. The paper provides explicit constructions, extensive error analysis, and numerical demonstrations on the Schrödinger and transport equations, illustrating the influence of on accuracy and cost. The framework both clarifies the link between splitting methods and exponential integrators for unbounded operators and points to feasible pathways for higher-order extensions using multidimensional Birkhoff quadratures.

Abstract

For linear differential equations of the form , , with a possibly unbounded operator , we construct and deduce error bounds for two families of second-order exponential splittings. The role of quadratures when integrating the twice-iterated Duhamel's formula is reformulated: we show that their choice defines the structure of the splitting. Furthermore, the reformulation allows us to consider quadratures based on the Birkhoff interpolation to obtain not only pure-stages splittings but also those containing derivatives of and commutators of and . In this approach, the construction and error analysis of the splittings are carried out simultaneously. We discuss the accuracy of the members of the families. Numerical experiments are presented to complement the theoretical consideration.
Paper Structure (18 sections, 2 theorems, 72 equations, 3 figures)

This paper contains 18 sections, 2 theorems, 72 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Family of Splittings $\mathcal{F}(h,\tau)$
  3. Derivation of the family of splittings
  4. First-order version
  5. Second-order version
  6. Error terms and their bounds
  7. Error terms $R_{V_1}$ and $R_{V_2}$
  8. Quadrature error terms: $R_{I_1}$ and $R_{I_2}$
  9. Accuracy of the family of integrators
  10. The role of parameter $\tau$
  11. Family of Splittings $\mathcal{D}(h,\tau)$
  12. Derivation of the family of splittings
  13. Error terms and their bounds
  14. The role of parameter $\tau$
  15. Numerical Examples
  16. ...and 3 more sections

Key Result

Theorem 2.4

Under Assumptions main_assum and commutators_assum.a, the family of integrators (family) satisfy meanwhile, under Assumptions main_assum and commutators_assum.b, it performs third (local) order of accuracy where constant $C_1$ depends on $\tau$, $C_h$ and $\|B'\|_Y$, while $C_2$ additionally depends on $\|B\|_Y$ and $\|B"\|_Y$.

Figures (3)

  • Figure 1: Distribution of the nodes in quadrature $I_2$ over the 2-d simplex (triangle), see (\ref{['quadrature2']}). At $\tau=0$ they lie on the vertices of the triangle. Meanwhile, at $\tau=1/2$ they all coincide at $(h/2,h/2)$. For both cases, they define a boundary-type quadrature and result in splitting featuring three exponentials. For $0<\tau<1/2$, we arrive at methods composed of five exponentials, and the nodes lie as displayed in the figure.
  • Figure 2: Global Error (in log-log scale) as a function of the time step $h$ for representative values of $\tau$. The black-dashed line represents the plot of $h^2$. The curves for $\tau=0.21,0.25$, see green curves, overlap each other.
  • Figure 3: Global Error (in log-log scale) as a function of the time step $h$ for representative values of $\tau$. The black-dashed line represents the plot of $h^2$. The inset shows how $\tau=1/2$ leads to slightly better results than $\tau=0.4,0.8$.

Theorems & Definitions (5)

  • Remark 2.2
  • Theorem 2.4
  • Theorem 3.2
  • Remark 4.1
  • Remark 4.2