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MLS-Based Selective Limiting for Shallow Waters Equations

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Part of the Springer Tracts in Mechanical Engineering book series (STME)

Abstract

Finite Volume Methods were successfully used in the last years to solve differential hyperbolic problems (Leveque, Finite-volume methods for hyperbolic problems. Cambridge University Press, Cambridge, 2005). Our research is focused here on the use of the Moving Least Squares (MLS) approximations for the development of a selective limiting technique to keep the accuracy of high-order methods in non-smooth flows. Following (Nogueira et al., Comput Methods Appl Mech Eng 199:2544–2558, 2010) we use the multiresolution properties of the MLS methodology and we define a shock-detection technique to act as a smoothness indicator. This sensor is used to detect shock waves present in the flow problem. The use of this technique combined with slope limiters improves the accuracy of the resulting TVD scheme. In this work we present the first results obtained with this technique applied to the resolution of the shallow waters equations with a high-order FV-MLS scheme (Cueto-Felgueroso et al., Comput Methods Appl Mech Eng 196:4712–4736, 2007). We present several 1D results and we compare them with those obtained with other high-order schemes.

Keywords

  • Shallow Water Equation
  • Move Little Square
  • Finite Volume Scheme
  • Slope Limiter
  • Smoothing Length

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Cueto-Felgueroso L, Colominas I, Fe J, Navarrina F, Casteleiro M (2006) High order finite volume schemes on unstructured grids using Moving Least Squares reconstruction. Application to shallow water dynamics. Int J Numer Methods Eng 65:295–331

    MATH  MathSciNet  CrossRef  Google Scholar 

  2. Cueto-Felgueroso L, Colominas I, Nogueira X, Navarrina F, Casteleiro M (2007) Finite volume solvers and Moving Least-Squares approximations for the compressible Navier-Stokes equations on unstructured grids. Comput Methods Appl Mech Eng 196:4712–4736

    MATH  MathSciNet  CrossRef  Google Scholar 

  3. Daubechies I (1992) Ten lectures on wavelets. Society for Industrial and Applied Mathematics, Philadelphia

    CrossRef  Google Scholar 

  4. Douglas CA, Harrison GP, Chick JP (2008) Life cycle assessment of the Seagen marine current turbine. Proc Inst Mech Eng M J Eng Marit Environ 222(1):1–12

    Google Scholar 

  5. Gossler A (2001) Moving least-squares: a numerical differentiation method for irregularly spaced calculation points. SANDIA Report, SAND2001-1669

    Google Scholar 

  6. Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 155(37):141–158

    MathSciNet  CrossRef  Google Scholar 

  7. Leveque RJ (1990) Numerical methods for conservation laws. Birkhauser, Basel

    MATH  CrossRef  Google Scholar 

  8. Leveque RJ (2005) Finite-volume methods for hyperbolic problems. Cambridge University Press, Cambridge

    Google Scholar 

  9. Nogueira X, Cueto-Felgueroso L, Colominas I, Khelladi S (2010) On the simulation of wave propagation with a higher order finite volume scheme based in Reproducing Kernel methods. Comput Methods Appl Mech Eng 199(155):1471–1490

    MATH  MathSciNet  CrossRef  Google Scholar 

  10. Nogueira X, Cueto-Felgueroso L, Colominas I, Navarrina F, Casteleiro M (2010) A new shock-capturing technique based on Moving Least Squares for higher-order numerical schemes on unstructured grids. Comput Methods Appl Mech Eng 199:2544–2558

    MATH  MathSciNet  CrossRef  Google Scholar 

  11. Toro EF (1999) Riemann solvers and numerical methods for fluid dynamics, 2nd edn. Springer, Heidelberg

    MATH  CrossRef  Google Scholar 

  12. Toro EF (2001) Shock-capturing methods for free-surface shallow flows. Manchester Metropolitan University

    MATH  Google Scholar 

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Acknowledgements

This work has been partially supported by the Ministerio de Ciencia e Innovación (#DPI2010-16496) and the Ministerio de Economía y Competitividad (#DPI2012-33622) of the Spanish Government and by the Consellería de Cultura, Educación e Ordenación Universitaria of the Xunta de Galicia (grant # GRC2014/039 ) cofinanced with FEDER funds, and the Universidade da Coruña.

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Cernadas, J., Nogueira, X., Colominas, I. (2015). MLS-Based Selective Limiting for Shallow Waters Equations. In: Ferrer, E., Montlaur, A. (eds) CFD for Wind and Tidal Offshore Turbines. Springer Tracts in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-16202-7_10

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  • DOI: https://doi.org/10.1007/978-3-319-16202-7_10

  • Publisher Name: Springer, Cham

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