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A pair of Second-order complex-valued, N-split operator-splitting methods

Raymond J. Spiteri, Siqi Wei

TL;DR

This work generalizes operator-splitting to $N$-split problems and introduces two second-order, complex-valued base methods with positive real parts (conjugate pairs) that are computationally inexpensive. A generalized Baker–Campbell–Hausdorff framework for $N$-split products is developed, yielding explicit order conditions up to $2$ and guiding the construction of the new methods. The authors derive two-stage complex Lie–Trotter (CLT-2) methods and demonstrate via composition how higher-order schemes can be built, with numerical experiments showing favorable accuracy and efficiency on real and complex systems. For complex-valued ODEs, these methods can outperform real-valued counterparts, while Strang remains an efficient baseline for real-valued problems. The results offer practical, robust tools for multi-operator splitting across diverse applications, and pave the way for higher-order, positive-real-part $N$-split integrators.

Abstract

The use of operator-splitting methods to solve differential equations is widespread, but the methods are generally only defined for a given number of operators, most commonly two. Most operator-splitting methods are not generalizable to problems with $N$ operators for arbitrary $N$. In fact, there are only two known methods that can be applied to general $N$-split problems: the first-order Lie--Trotter (or Godunov) method and the second-order Strang (or Strang--Marchuk) method. In this paper, we derive two second-order operator-splitting methods that also generalize to $N$-split problems. These methods are complex valued but have positive real parts, giving them favorable stability properties, and require few sub-integrations per stage, making them computationally inexpensive. They can also be used as base methods from which to construct higher-order $N$-split operator-splitting methods with positive real parts. We verify the orders of accuracy of these new $N$-split methods and demonstrate their favorable efficiency properties against well-known real-valued operator-splitting methods on both real-valued and complex-valued differential equations.

A pair of Second-order complex-valued, N-split operator-splitting methods

TL;DR

This work generalizes operator-splitting to -split problems and introduces two second-order, complex-valued base methods with positive real parts (conjugate pairs) that are computationally inexpensive. A generalized Baker–Campbell–Hausdorff framework for -split products is developed, yielding explicit order conditions up to and guiding the construction of the new methods. The authors derive two-stage complex Lie–Trotter (CLT-2) methods and demonstrate via composition how higher-order schemes can be built, with numerical experiments showing favorable accuracy and efficiency on real and complex systems. For complex-valued ODEs, these methods can outperform real-valued counterparts, while Strang remains an efficient baseline for real-valued problems. The results offer practical, robust tools for multi-operator splitting across diverse applications, and pave the way for higher-order, positive-real-part -split integrators.

Abstract

The use of operator-splitting methods to solve differential equations is widespread, but the methods are generally only defined for a given number of operators, most commonly two. Most operator-splitting methods are not generalizable to problems with operators for arbitrary . In fact, there are only two known methods that can be applied to general -split problems: the first-order Lie--Trotter (or Godunov) method and the second-order Strang (or Strang--Marchuk) method. In this paper, we derive two second-order operator-splitting methods that also generalize to -split problems. These methods are complex valued but have positive real parts, giving them favorable stability properties, and require few sub-integrations per stage, making them computationally inexpensive. They can also be used as base methods from which to construct higher-order -split operator-splitting methods with positive real parts. We verify the orders of accuracy of these new -split methods and demonstrate their favorable efficiency properties against well-known real-valued operator-splitting methods on both real-valued and complex-valued differential equations.
Paper Structure (10 sections, 5 theorems, 50 equations, 4 figures, 4 tables)

This paper contains 10 sections, 5 theorems, 50 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. $N$-split operator-splitting methods
  3. Main results
  4. The BCH formula for $N$-split problems
  5. Order conditions via the generalized BCH formula
  6. Derivation of $N$-split operator-splitting methods
  7. Numerical experiments
  8. 2D ADR problem
  9. $3$-split complex-valued ODE
  10. Conclusions

Key Result

Lemma 2.1

[Gröbner 1960] Let $\varphi_s^{[1]}$ and $\varphi_t^{[2]}$ be the flows of the differential equations $\dv{y^{[1]}}{t} = \mathcal{F}^{[1]} (t, y^{[1]})$ and $\dv{y^{[2]}}{t} = \mathcal{F}^{[2]} (t, y^{[2]})$, respectively. For their composition, we then have where $D^{[\ell]} = \sum\limits_{j} \mathcal{F}^{[\ell]}_j \pdv{}{y_j}$ is the Lie derivative.

Figures (4)

  • Figure 1: Order of convergence of Strang, CLT-2, $\Psi^{\text{CLT-2}}_{\Delta t}(3)$, and $\Psi^{\text{S}}_{\Delta t}(3)$ applied to the 2D ADR problem \ref{['eq:ADR']}.
  • Figure 2: Work precision diagram of Strang, CLT-2, $\Psi^{\text{CLT-2}}_{\Delta t}(3)$, and $\Psi^{\text{S}}_{\Delta t}(3)$ applied to the 2D ADR problem \ref{['eq:ADR']}.
  • Figure 3: Convergence of Strang, CLT-2, $\Psi^{\text{CLT-2}}_{\Delta t}(3)$, $\Psi^{\text{S}}_{\Delta t}(3)$ and PP 3/4 A 3 applied to the complex ODE \ref{['eq:complex_ode']}.
  • Figure 4: Work-precision diagram of Strang, CLT-2, $\Psi^{\text{CLT-2}}_{\Delta t}(3)$, $\Psi^{\text{S}}_{\Delta t}(3)$, and PP 3/4 A 3 applied to the complex ODE \ref{['eq:complex_ode']} and the system of real ODEs \ref{['complex_ode_real']}.

Theorems & Definitions (15)

  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • ...and 5 more