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Partially Implicit Runge-Kutta Methods for Wave-Like Equations

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Advances in Differential Equations and Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 4))

Abstract

Runge-Kutta methods are used to integrate in time systems of differential equations. Implicit methods are designed to overcome numerical instabilities appearing during the evolution of a system of equations. We will present partially implicit Runge-Kutta methods for a particular structure of equations, generalization of a wave equation; the partially implicit term refers to this structure, where the implicit term appears only in a subset of the system of equations. These methods do not require any inversion of operators and the computational costs are similar to those of explicit Runge-Kutta methods. Partially implicit Runge-Kutta methods are derived up to third-order of convergence. We analyze their stability properties and show the practical applicability in several numerical examples.

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Acknowledgements

This work has been funded by the SN2NS project ANR-10-BLAN-0503, the Spanish MICINN (AYA 2010-21097-C03-01), the Generalitat Valenciana (PROMETEO-2009-103 and PROMETEO-2011-083) and the ERC Starting Grant CAMAP-259276.

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Authors and Affiliations

  1. Laboratoire Univers et Théories, CNRS/Laboratoire de Paris/Université Paris Diderot, 5 place Jules Janssen, F-92195, Meudon, France

    Isabel Cordero-Carrión

  2. Departamento de Astronomía y Astrofísica, Universidad de Valencia, C/Dr. Moliner, 50, E-46100, Valencia, Spain

    Pablo Cerdá-Durán

Authors
  1. Isabel Cordero-Carrión
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  2. Pablo Cerdá-Durán
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Corresponding author

Correspondence to Isabel Cordero-Carrión .

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Editors and Affiliations

  1. Dept. de Mathemàtiques and IMAC, Universitat Jaume I, Castelló, Spain

    Fernando Casas

  2. Dept. de Mathemàtiques and IMAC, Universitat Jaume I, Castelló, Spain

    Vicente Martínez

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Cordero-Carrión, I., Cerdá-Durán, P. (2014). Partially Implicit Runge-Kutta Methods for Wave-Like Equations. In: Casas, F., Martínez, V. (eds) Advances in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-06953-1_26

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