Computing the Dynamics of Complex Singularities of Nonlinear PDEs

A two-step strategy is proposed for the computation of singularities in nonlinear PDEs. The first step is the numerical solution of the PDE using a Fourier spectral method; the second step involves numerical analytical continuation into the complex plane using the epsilon algorithm to sum the Fourier series. Test examples include the inviscid Burgers and nonlinear heat equations, as well as a transport equation involving the Hilbert transform. Numerical results, including Web animations that show the dynamics of the singularities in the complex plane, are presented.