Numerical Experiments on a Nonlinear Wave Equation with Singular Solutions

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 Holder 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.