A pseudo-spectral splitting method for linear dispersive problems with transparent boundary conditions

A pseudo-spectral Strang splitting method for linear dispersive problems with transparent boundary conditions

“…In this section we derive a semi-discrete scheme by applying the Strang splitting method to problem (1) restricted to a finite interval [a, b], where a < b. Inspired by the ideas in [12], we perform a time splitting in order to separate the advection problem…”

“…We notice that the cross marks representing the exact solution lie on the numerical solution In Table 2 the full discretization error between the numerical solution and the exact solution varying N and M is reported. In particular, in Table 2 (left) the number of time steps M is fixed to 2 12 and the number of collocation points N is varying from 24 to 40. In this way the time discretization error is small enough to be negligible with respect to the spatial error.…”

“…The goal of this work is to design a splitting scheme that is second order in time with spectral accuracy in space. This paper can be seen as an extension to [12], where a splitting scheme of order one in time and spectral accuracy in space is presented. When solving (1) one of the main difficulties one has to face is the unbounded domain R. Numerical simulations typically consider a finite domain that leads to boundary conditions.…”

“…A lot of work has been done for the Schrödinger equation in recent years, see [1,3,5] and references therein. For third-order problems, we refer the reader to [6,7,12,23] and references therein.…”

“…We therefore employ a time-splitting approach in order to separate the advection problem from the dispersive problem. Operator splitting methods for dispersive problems have been employed and studied before, we refer the reader to [9,11,12,15]. For splitting method with absorbing boundary conditions we cite the work [5].…”