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Operator-Splitting on Hyperbolic Balance Laws

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Advances in Differential Equations and Applications

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

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Abstract

Operator-Splitting Methods or Fractional-Step Methods are based on the fact that hyperbolic balance laws can be split exactly into a homogeneous hyperbolic partial differential equation (PDE, advection) and an ordinary differential equation (EDO, evolution); which means that both advecting first and evolving next and evolving first and advecting next are equivalent to solve the whole problem directly [3, 4, 10, 11]. The key to this method is the physical flux which must be used in the advective part. If the problem is linear, it coincides with the physical flux of the problem and does not depend on whether the advection is solved before or after the evolution. However, this is no longer true for nonlinear problems: it is different from the flux of the problem and depends on the order [8]. In this work we will begin with the analysis of the splitting of multi-dimensional linear systems and we will end up explaining how exact nonlinear splitting can be obtained for one-dimensional scalar equations.

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Acknowledgements

The authors are indebted to Professor E. F. Toro for many valuable discussions. This work was financially supported by Spanish MICINN project CGL2011-28499-C03-01.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Santiago de Compostela, Lope Gómez de Marzoa s/n Campus sur, 15782, Santiago de Compostela, Spain

    Pedro González de Alaiza Martínez & María Elena Vázquez-Cendón

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  1. Pedro González de Alaiza Martínez
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  2. María Elena Vázquez-Cendón
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Correspondence to Pedro González de Alaiza Martínez .

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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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González de Alaiza Martínez, P., Vázquez-Cendón, M.E. (2014). Operator-Splitting on Hyperbolic Balance Laws. 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_27

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