A Schwarz waveform relaxation method for time-dependent space fractional Schrödinger/heat equations (Q2085662)

The numerical solution of a linear time-dependent space fractional Schrödinger equation is discussed and analysed. For the approximation of the fractional Laplace operator a method is used which is presented in [\textit{Y. Huang} and \textit{A. Oberman}, SIAM J. Numer. Anal. 52, No. 6, 3056--3084 (2014; Zbl 1316.65071)]. The presented scheme preserves both mass and energy. The order of convergence of this scheme is illustrated by a numerical example. A non-overlapping Schwarz waveform relaxation domain decomposition method (SWR DDM) for solving the considered problem is described and analysed. Some numerical examples are given, which illustrate the convergence behavior of the method. Hereby, an extension of the proposed method to a nonlinear fractional Schrödinger equation is also given. Furthermore, it is discussed how the presented method can be extended to fractional heat equations.