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Numerical Experiments on a Nonlinear Wave Equation with Singular Solutions

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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 119))

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Abstract

We use a Fourier pseudospectral method to compute solutions to the Cauchy problem for a nonlinear variational wave equation originally proposed as a model for the dynamics of nematic liquid crystals. The solution is known to form singularities in finite time; in particular space and time derivatives become unbounded. Beyond the singularity time, both conservative and dissipative Hölder continuous weak solutions exist. We present results with energy-conserving discretizations as well as with a vanishing viscosity sequence, noting marked differences between the computed solutions after the solution loses regularity.

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Acknowledgements

This work was supported in part by NSF Grant DMS-1418871. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. We also acknowledge the use of FFT routines adapted from [7] as well as OpenMP-enabled FFTs adapted from [16] and written by Burkardt [5].

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Authors and Affiliations

  1. Southern Methodist University, Dallas, TX, USA

    Thomas Hagstrom

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  1. Thomas Hagstrom
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Correspondence to Thomas Hagstrom .

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Editors and Affiliations

  1. School of Mechanical Engineering, University of Campinas, Campinas, Brazil

    Marco L. Bittencourt

  2. Department of Civil Engineering, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil

    Ney A. Dumont

  3. EPFL-SB-MATHICSE-MCSS, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Jan S. Hesthaven

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Hagstrom, T. (2017). Numerical Experiments on a Nonlinear Wave Equation with Singular Solutions. In: Bittencourt, M., Dumont, N., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-319-65870-4_34

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