Details of Research Outputs

We propose an exponential spectral process (ESP) method for time discretization of spatial-temporal equations. The proposed ESP method uses explicit iterations at each time step, which allows us to use simple initializations at each iteration. This method has the capacity to obtain high accuracy (up to machine precision) with reasonably large time step sizes. Theoretically, the ESP method is proved to be unconditionally energy stable for arbitrary iteration steps for the case of using two spectral points. To demonstrate the advantages of the ESP method, we consider two applications which have stability difficulties in large time simulations. One of them is the Allen-Cahn equation with the symmetry breaking problem faced by most of the existing time discretizations, and the second one is about the complex Ginzburg-Landau equation which also suffers from large time instability.