Role of shift constant in Energy Shifted SAV for Hamiltonian Systems

Fabiana Zama, Michele Ducceschi, Stefan Bilbao

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract / Description of output

In recent years, considerable work has been devoted to the design of energy-stable numerical methods, a class of geometrical integration technique in which the discrete energy or pseudo-energy remains conserved. This property finds practical applications in systems for which the energy is bounded from below since the growth of the solutions is then itself bounded, yielding a form of stability. Such a property may be further exploited to transform the energy overall into a quadratic form, representing the core of recent numerical techniques such as the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV) approaches. These methods have been applied to a large class of problems due to their remarkable efficiency. Yet, several aspects of such techniques have seen little investigation. In this work, the role of the ``shift'' constant in the expression of the potential energy in Hamiltonian systems is investigated. This is a positive gauge constant that may be used to augment the expression of the energy. Since Hamilton's equations are given in terms of gradients of the Hamiltonian, such a constant has no influence on the resulting dynamics in the continuous system. However, empirical evidence has suggested that the convergence properties of the associated SAV approaches are affected by the magnitude of the shift factor. In this work, the behaviour of the relative global error of SAV schemes in the cases of the harmonic and Duffing oscillators is numerically investigated. The results reveal an optimal shift factor that increases the convergence rate. Using the optimal shift factor, the proposed method displays variable order accuracy ranging from second to twelfth order.
Original languageEnglish
Title of host publicationProceedings of the 12th International Conference on Mathematical Modeling in Physical Sciences
Place of PublicationBelgrade, Serbia
PublisherIOP Publishing
Number of pages10
Publication statusAccepted/In press - 7 Aug 2023
Event12th International Conference on Mathematical Modeling in Physical Sciences - Belgrade, Serbia
Duration: 28 Aug 202331 Aug 2023

Publication series

NameJournal of Physics: Conference Series
PublisherInstitute of Physics
ISSN (Print)1742-6596

Conference

Conference12th International Conference on Mathematical Modeling in Physical Sciences
Country/TerritorySerbia
CityBelgrade
Period28/08/2331/08/23

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