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Runge–Kutta–Möbius methods

Molnár, András ; Fekete, Imre LU and Söderlind, Gustaf LU (2023) In Periodica Mathematica Hungarica 87(1). p.167-181
Abstract

In the numerical integration of nonlinear autonomous initial value problems, the computational process depends on the step size scaled vector field hf as a distinct entity. This paper considers a parameterized transformation hf↦hf∘(I-γhf)-1,and its role in the finite step size stability of singly diagonally implicit Runge—Kutta (SDIRK) methods. For a suitably chosen γ> 0 , the transformed map is Lipschitz continuous with a reasonably small constant, whenever hf is negative monotone. With this transformation, an SDIRK method is equivalent to an explicit Runge–Kutta (ERK) method applied to the transformed vector field. Through this mapping, the SDIRK methods’ A-stability, and linear order conditions are investigated. The latter are... (More)

In the numerical integration of nonlinear autonomous initial value problems, the computational process depends on the step size scaled vector field hf as a distinct entity. This paper considers a parameterized transformation hf↦hf∘(I-γhf)-1,and its role in the finite step size stability of singly diagonally implicit Runge—Kutta (SDIRK) methods. For a suitably chosen γ> 0 , the transformed map is Lipschitz continuous with a reasonably small constant, whenever hf is negative monotone. With this transformation, an SDIRK method is equivalent to an explicit Runge–Kutta (ERK) method applied to the transformed vector field. Through this mapping, the SDIRK methods’ A-stability, and linear order conditions are investigated. The latter are closely related to approximations of the exponential function e z that are polynomial in z, when considering ERK methods, and polynomial in terms of the transformed variable z(1 - γz) - 1, in case of SDIRK methods. Considering the second family of methods, and expanding the exponential function in terms of this transformed variable, the coefficients can be expressed in terms of Laguerre polynomials. Lastly, a family of methods is constructed using the transformed vector field, and its order conditions, A-stability, and B-stability are investigated.

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author
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organization
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type
Contribution to journal
publication status
published
subject
keywords
Dissipative, Laguerre polynomial, Möbius transformation, Runge–Kutta method, SDIRK method, Stiff
in
Periodica Mathematica Hungarica
volume
87
issue
1
pages
167 - 181
publisher
Springer
external identifiers
  • scopus:85145739492
ISSN
0031-5303
DOI
10.1007/s10998-022-00510-5
language
English
LU publication?
yes
id
abfe3f42-1886-4774-8b63-659f425dd8b7
date added to LUP
2023-02-21 10:11:46
date last changed
2023-10-26 14:50:56
@article{abfe3f42-1886-4774-8b63-659f425dd8b7,
  abstract     = {{<p>In the numerical integration of nonlinear autonomous initial value problems, the computational process depends on the step size scaled vector field hf as a distinct entity. This paper considers a parameterized transformation hf↦hf∘(I-γhf)-1,and its role in the finite step size stability of singly diagonally implicit Runge—Kutta (SDIRK) methods. For a suitably chosen γ&gt; 0 , the transformed map is Lipschitz continuous with a reasonably small constant, whenever hf is negative monotone. With this transformation, an SDIRK method is equivalent to an explicit Runge–Kutta (ERK) method applied to the transformed vector field. Through this mapping, the SDIRK methods’ A-stability, and linear order conditions are investigated. The latter are closely related to approximations of the exponential function e <sup>z</sup> that are polynomial in z, when considering ERK methods, and polynomial in terms of the transformed variable z(1 - γz) <sup>- 1</sup>, in case of SDIRK methods. Considering the second family of methods, and expanding the exponential function in terms of this transformed variable, the coefficients can be expressed in terms of Laguerre polynomials. Lastly, a family of methods is constructed using the transformed vector field, and its order conditions, A-stability, and B-stability are investigated.</p>}},
  author       = {{Molnár, András and Fekete, Imre and Söderlind, Gustaf}},
  issn         = {{0031-5303}},
  keywords     = {{Dissipative; Laguerre polynomial; Möbius transformation; Runge–Kutta method; SDIRK method; Stiff}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{167--181}},
  publisher    = {{Springer}},
  series       = {{Periodica Mathematica Hungarica}},
  title        = {{Runge–Kutta–Möbius methods}},
  url          = {{http://dx.doi.org/10.1007/s10998-022-00510-5}},
  doi          = {{10.1007/s10998-022-00510-5}},
  volume       = {{87}},
  year         = {{2023}},
}