Numerical Experiments on a Nonlinear Wave Equation with Singular Solutions
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.
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  as well as OpenMP-enabled FFTs adapted from  and written by Burkardt .
- 1.P. Aursand. U. Koley, Local discontinuous Galerkin schemes for a nonlinear variational wave equation modeling liquid crystals. Preprint (2014)Google Scholar
- 5.J. Burkardt, http://www.sc.fsu.edu/~jburkardt/
- 9.E. Hairer, http://www.unige.ch/~hairer/software.html
- 17.N. Yi, H. Liu, An energy conserving discontinuous Galerkin method for a nonlinear variational wave equation. Commun. Comput. Phys. (2016, Preprint)Google Scholar