| Abstract: |
| 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 has been shown to be unconditionally energy stable for arbitrary number of iteration steps for the case where two spectral points are used. To demonstrate the advantages of the ESP approach, we consider two applications that have stability difficulties in large-time simulations. One of them is the Allen-Cahn equation with the symmetry breaking problem that most existing time discretizations face, and the second one is about the complex Ginzburg-Landau equation, which also suffers from large-time instabilities. |
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