Analysis of a third-order absorbing boundary condition for the Schrödinger equation discretized in space.
A Schrödinger type equation of the form \[ \partial_t u = -{i \over c} ( \partial^2_x u +V u) \] for \(x\in \mathbb R,\;t>0\) and with \(c>0, V \in \mathbb R\) is considered especially from the numerical point of view. For obtaining the reliable numerical solution by space discretization a finite spatial subdomain must be used with appropriate boundary conditions. One of the wide spread techniques is to use local boundary conditions (ABC) obtained by approximation the transparent, or reflection free, boundary conditions (TBC). ABC boundary conditions are obtained by interpolatory techniques and usually are denoted as \(ABC(j_1,j_2)\) when \(j_1+j_2+1\) interpolatory nodes are used. It was proved in previous papers that the matrices associated with \(ABC(1,0)\) as well \(ABC(1,1)\) have all eigenvalues with negative real part and the semidiscrete problem is weak ill-posed when the spatial step size \(h\) goes to zero. A semidiscrete alternative to these ABC was developed denoted by \(SABC(j_1,j_2)\). Similar properties as \(ABC(1,1)\) for \(SABC(1,1)\), that means weakly ill-posed, are proved in this paper. The essential difference is in the possibility to use bigger values of the spatial discretiazation in order to obtain a certain absorption, what is important because of the weak instability of both problems. Stability of the matrix for the resulting semidiscrete problem obtained for \(SABC(1,1)\) is proved and a bound for the possible growth of the norm of the solution is obtained.
scientific article; zbMATH DE number 2138569