A class of explicit multistep exponential integrators for semilinear problems

Calvo, M. P.; Palencia, C.

doi:10.1007/s00211-005-0627-0

Cited by 86 publications

(95 citation statements)

References 24 publications

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“…Motivated by the approach in [1], we suggest to construct starting values by replacing the nonlinearity g n in (3.3) by the interpolation polynomial…”

Section: Linearized Exponential Multistep Methodsmentioning

confidence: 99%

“…The main contribution of our paper is a rigorous error analysis for these methods. Note that Calvo and Palencia [1] constructed and analyzed a related class of k-step methods, where the variation-ofconstants formula is taken over an interval of length kh instead of h. In contrast to exponential Adams methods, all parasitic roots of their methods are on the unit circle.…”

Section: Introductionmentioning

confidence: 99%

“…We present here a general class of linearized exponential multistep methods which is based on interpolation with a Hermite node at the beginning of each step. For both kind of methods, we propose starting procedures which are inspired by a construction in [1].…”

Section: Introductionmentioning

confidence: 99%

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Exponential multistep methods of Adams-type

Hochbruck

Ostermann

2011

Bit Numer Math

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The paper is concerned with the construction, implementation and numerical analysis of exponential multistep methods. These methods are related to explicit Adams methods but, in contrast to the latter, make direct use of the exponential and related matrix functions of a (possibly rough) linearization of the vector field. This feature enables them to integrate stiff problems explicitly in time.A stiff error analysis is performed in an abstract framework of linear semigroups that includes semilinear evolution equations and their spatial discretizations. A possible implementation of the proposed methods, including the computation of starting values and the evaluation of the arising matrix functions by Krylov subspace methods is discussed. Moreover, an interesting connection between exponential Adams methods and a class of local time stepping schemes is established.Numerical examples that illustrate the methods' properties are included.

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“…Motivated by the approach in [1], we suggest to construct starting values by replacing the nonlinearity g n in (3.3) by the interpolation polynomial…”

Section: Linearized Exponential Multistep Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exponential multistep methods of Adams-type

Hochbruck

Ostermann

2011

Bit Numer Math

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“…Cox and Matthews (2002) consider the same class of methods, while Calvo and Palencia (2006) constructed and analysed k-step methods, where the variation-of-constants formula is taken over an interval of length kh instead of h. In contrast to Adams methods, their methods have all parasitic roots on the unit circle. A variety of explicit and implicit schemes is given in Beylkin, Keiser and Vozovoi (1998).…”

Section: Exponential Multistep Methodsmentioning

confidence: 99%

Exponential Integrators

Ostermann

2015

Encyclopedia of Applied and Computational Mathematics

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In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system.Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in which exponential integrators are used.

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“…This was the reason why, until recently, these methods have not been regarded as practical. The latest achievements in the field of computing approximations to the matrix exponential, have provided a new interest in the construction and implementation of exponential integrators [2,3,6,7,9].…”

Section: Introductionmentioning

confidence: 99%

Integrating Factor Methods as Exponential Integrators

Minchev¹

2006

Large-Scale Scientific Computing

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Abstract. Recently a lot of effort has been placed in the construction and implementation of a class of methods called exponential integrators. These methods are preferable when one has to deal with stiff and highly oscillatory semilinear problems, which often arise after spatial discretization of Partial Differential Equations (PDEs). The main idea behind the methods is to use the exponential and some closely related functions inside the numerical scheme. In this note we show that the integrating factor methods, introduced by Lawson in 1967, are also examples of exponential integrators with very special structure for the related exponential functions. In order to prove this relation, we use the approach based on bi-coloured rooted trees and B-series. We also show under what conditions every bi-coloured rooted tree can be express as a linear combination of standard non-coloured rooted trees.

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A class of explicit multistep exponential integrators for semilinear problems

Cited by 86 publications

References 24 publications

Exponential multistep methods of Adams-type

Exponential multistep methods of Adams-type

Exponential Integrators

Integrating Factor Methods as Exponential Integrators

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