A Fourier-Chebyshev pseudospectral methods for the investigation of potential singularities in the axisymmetric 3D Euler equation*

We present a Fourier-Chebyshev pseudospectral scheme we to investigate the potential singularity in the axisymmetric 3D Euler equation (cf.  G. Luo and T. Hou, Proc. Natl. Acad. Sci. USA 111, 12968 (2014). We demonstrate that: (a) the singularity time is preceded in pseudospectral simulations bythe formation of localised oscillatory structures called tygers; and  (b)  a generalisation of the analyticity-strip method  can be used to track the movement of the complex singularities of the velocity field, from which we can get an estimate for the singularity time. We explore generalizations of this method to other ideal hydrodynamical equations.