## B-series and order conditions for exponential integrators

Citations: | 10 - 4 self |

### BibTeX

@TECHREPORT{Owren_b-seriesand,

author = {Brynjulf Owren and B˚ard Skaflestad},

title = {B-series and order conditions for exponential integrators},

institution = {},

year = {}

}

### OpenURL

### Abstract

Abstract. We introduce a general format of numerical ODE-solvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of B-series and bicolored rooted trees. To ease the construction of specific schemes we generalize an idea of Zennaro [Math. Comp., 46 (1986), pp. 119–133] and define natural continuous extensions in the context of exponential integrators. This leads to a relatively easy derivation of some of the most popular recently proposed schemes. The general format of schemes considered here makes use of coefficient functions which will usually be selected from some finite dimensional function spaces. We will derive lower bounds for the dimension of these spaces in terms of the order of the resulting schemes. Finally, we illustrate the presented ideas by giving examples of new exponential integrators of orders 4 and 5.

### Citations

516 |
Solving ordinary differential equations
- HAIRER, WANNER
- 1991
(Show Context)
Citation Context ...y write T ∪∅= ⋃ W m + (Tb ∪∅), Tw = ⋃ W m (2.3) + (Tb ∪∅). m≥0 The same bicolored trees used here also appear in the linearly implicit W -methods; see Steihaug and Wolfbrandt [21] as well as the text =-=[10]-=- by Hairer and Wanner. Following, for instance, the text by Hairer, Lubich, and Wanner [8], we may work with formal B-series. For an arbitrary map c : T ∪∅→R, we let the formal series (2.4) B(c,u)=c(∅... |

425 |
Solving Ordinary Differential Equations I: Nonstiff Problems
- Hairer, Norsett, et al.
- 2000
(Show Context)
Citation Context ...the type (1.1). We will use the We will discuss conditions on the coefficients α j,k rORDER CONDITIONS FOR EXPONENTIAL INTEGRATORS 1717 well known approach involving rooted trees; see, for instance, =-=[9, 2]-=-. The conditions we find will depend only on the first αj,k r for k ≤ p − 2 and on βj,k for k ≤ p − 1. On this note we will not address issues related to the behavior of the coefficient functions aj r... |

287 |
Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations
- Hairer, Lubich, et al.
- 2006
(Show Context)
Citation Context ...d here also appear in the linearly implicit W -methods; see Steihaug and Wolfbrandt [21] as well as the text [10] by Hairer and Wanner. Following, for instance, the text by Hairer, Lubich, and Wanner =-=[8]-=-, we may work with formal B-series. For an arbitrary map c : T ∪∅→R, we let the formal series (2.4) B(c,u)=c(∅)u + ∑ τ∈T m≥1 h |τ| c(τ)F (τ)(u) σ(τ) be a B-series, where σ(τ) is the symmetry coefficie... |

149 |
Numerical Methods for Ordinary Differential Equations
- Butcher
- 2003
(Show Context)
Citation Context ...the type (1.1). We will use the We will discuss conditions on the coefficients α j,k rORDER CONDITIONS FOR EXPONENTIAL INTEGRATORS 1717 well known approach involving rooted trees; see, for instance, =-=[9, 2]-=-. The conditions we find will depend only on the first αj,k r for k ≤ p − 2 and on βj,k for k ≤ p − 1. On this note we will not address issues related to the behavior of the coefficient functions aj r... |

98 | Exponential integrators for large systems of differential equations
- Hochbruck, Lubich, et al.
- 1998
(Show Context)
Citation Context ...n [6], and Friedli [7] to mention just a few. Recently there has been a revived interest in these schemes, in particular for the solution of nonlinear partial differential equations; see for instance =-=[11, 17, 5, 3, 14, 13]-=-. For a thorough review of the history of exponential integrators; see [16] and the references therein. The integrators found in these papers are derived in rather different ways, and they are formula... |

83 |
Exponential time differencing for stiff systems
- Cox, Matthews
- 2002
(Show Context)
Citation Context ...n [6], and Friedli [7] to mention just a few. Recently there has been a revived interest in these schemes, in particular for the solution of nonlinear partial differential equations; see for instance =-=[11, 17, 5, 3, 14, 13]-=-. For a thorough review of the history of exponential integrators; see [16] and the references therein. The integrators found in these papers are derived in rather different ways, and they are formula... |

56 | High order Runge–Kutta methods on manifolds
- Munthe-Kaas
- 1999
(Show Context)
Citation Context ...n [6], and Friedli [7] to mention just a few. Recently there has been a revived interest in these schemes, in particular for the solution of nonlinear partial differential equations; see for instance =-=[11, 17, 5, 3, 14, 13]-=-. For a thorough review of the history of exponential integrators; see [16] and the references therein. The integrators found in these papers are derived in rather different ways, and they are formula... |

54 | Owren B.: Computations in a free Lie algebra
- Munthe-Kaas
- 1998
(Show Context)
Citation Context ...2) z 4 φ0(z/2) 2 φ0(z/2) 1 2 φ0(z) − 1 3 φ0(z/2) 1 3 φ0(z) 1 3 φ0(z) − 1 1 φ0(z)+ 6 3 φ0(z/2) Table 1 gives the coefficient functions aj r(z) and br (z) for the fourth order RKMK scheme introduced in =-=[18]-=- in this general format when applied to the problem (1.1) with an affine Lie group action, and the commutator-free scheme of order 4 from [3]; in both tables φ0(z) =(ez−1)/z. For deriving order condit... |

43 | Fourth-Order Time Stepping for Stiff PDEs
- Trefethen
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Citation Context |

31 |
Natural continuous extensions of Runge-Kutta methods
- Zennaro
- 1986
(Show Context)
Citation Context ...ox and Matthews [5] presented a fourth order scheme using these basis functions with Kb = 3. Krogstad [14] also derived a variant of their method by using a continuous extension as just explained. In =-=[22]-=- Zennaro developed a theory which generalizes the collocation polynomial idea to arbitrary Runge–Kutta schemes. The approach was called natural continuous extensions (NCE). By making a slight modifica... |

30 | Explicit exponential Runge-Kutta methods for semilinear parabolic problems
- Hochbruck, Ostermann
(Show Context)
Citation Context ...k r for k ≤ p − 2 and on βj,k for k ≤ p − 1. On this note we will not address issues related to the behavior of the coefficient functions aj r(z) and br (z) for large values of z. In the recent paper =-=[12]-=-, an order theory for explicit exponential integrators is presented and its application to semilinear parabolic problems is discussed. While classical or nonstiff order conditions are usually derived ... |

26 |
Generalized integrating factor methods for stiff PDEs
- Krogstad
- 2005
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24 | Generalized Runge-Kutta processes for stable systems with large Lipschitz constants - Lawson - 1967 |

17 | The solution of ordinary differential equations with large time constants - Certaine - 1960 |

17 |
Order conditions for numerical integrators obtained by composing simpler integrators
- Murua, Sanz-Serna
- 1999
(Show Context)
Citation Context ...ng of all trees in T except those with a terminal white node. There is an interesting connection between the set of trees T ′ and the trees used to develop the order theory for composition methods in =-=[19]-=-. White nodes appear as connected strings of nodes which, except for the root, have exactly one parent and one child, and always terminate in a black node. Therefore one can remove all white nodes and... |

16 |
An Attempt to Avoid Exact Jacobian and Nonlinear
- Steihaug, Wolfbrandt
- 1979
(Show Context)
Citation Context ...+(∅) = •, W+(∅) =◦, we may write T ∪∅= ⋃ W m + (Tb ∪∅), Tw = ⋃ W m (2.3) + (Tb ∪∅). m≥0 The same bicolored trees used here also appear in the linearly implicit W -methods; see Steihaug and Wolfbrandt =-=[21]-=- as well as the text [10] by Hairer and Wanner. Following, for instance, the text by Hairer, Lubich, and Wanner [8], we may work with formal B-series. For an arbitrary map c : T ∪∅→R, we let the forma... |

14 |
Exponential integrators for semilinear problems
- Minchev
- 2004
(Show Context)
Citation Context ...hese schemes, in particular for the solution of nonlinear partial differential equations; see for instance [11, 17, 5, 3, 14, 13]. For a thorough review of the history of exponential integrators; see =-=[16]-=- and the references therein. The integrators found in these papers are derived in rather different ways, and they are formulated for different types of systems of differential equations. On this note,... |

14 |
An A-stable modification of the Adams-Bashforth methods
- Norsett
- 1969
(Show Context)
Citation Context ...M99 DOI. 10.1137/040612683 1. Introduction. Numerical integration schemes which use the matrix exponential go back all the way to Certaine [4], but there are also early papers by Lawson [15], Nørsett =-=[20]-=-, Ehle and Lawson [6], and Friedli [7] to mention just a few. Recently there has been a revived interest in these schemes, in particular for the solution of nonlinear partial differential equations; s... |

12 | Commutator-free Lie group methods
- Celledoni, Marthinsen, et al.
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Citation Context |

12 |
Verallgemeinerte Runge-Kutta Verfahren zur LÄosung steifer Di®erentialgleichungssysteme, in Numerical Treatment of Di®erential Equations
- Friedli
- 1978
(Show Context)
Citation Context ...ion. Numerical integration schemes which use the matrix exponential go back all the way to Certaine [4], but there are also early papers by Lawson [15], Nørsett [20], Ehle and Lawson [6], and Friedli =-=[7]-=- to mention just a few. Recently there has been a revived interest in these schemes, in particular for the solution of nonlinear partial differential equations; see for instance [11, 17, 5, 3, 14, 13]... |

8 | Generalized Runge–Kutta processes for stiff initial-value problems - Ehle, Lawson - 1975 |