Runge–Kutta–Möbius methods

András Molnár, Imre Fekete, Gustaf Söderlind

Research output: Contribution to journalArticlepeer-review

Abstract (may include machine translation)

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.

Original languageEnglish
Pages (from-to)167-181
Number of pages15
JournalPeriodica Mathematica Hungarica
Volume87
Issue number1
DOIs
StatePublished - 5 Jan 2023
Externally publishedYes

Keywords

  • Dissipative
  • Laguerre polynomial
  • Möbius transformation
  • Runge–Kutta method
  • SDIRK method
  • Stiff

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