Discrete Absorbing Boundary Conditions for Schrödinger-Type Equations. Construction and Error Analysis

Abstract

Abstract. Recently, some absorbing boundary conditions for Schrödinger-type equations have been studied by Fevens, Jiang and Alonso-Mallo, and Reguera. These conditions make it possible to obtain a very high absorption at the boundary avoiding the nonlocality of transparent boundary conditions. However, the implementations used in the literature, where the boundary condition is chosen in a manual way in accordance with the solution or fixed independently of the solution, are not practical because of the small absorption. In this paper, a new practical adaptive implementation is developed that allows us to obtain automatically a very high absorption.

A proof of the well posedness of discretized wave equation with an absorbing boundary condition

“…An asymptotic stability is obtained for the smallest orders of absorption, taking as main parameter the ratio between the time step size and the parameter of the spatial discretization. We now prove that a similar result to the one obtained in [4,5] for Schrö-dinger-type equations is true for spatially discrete wave equations for an ABC with fifth order of absorption, obtained with the technique introduced in [17], for one dimensional wave equations previously discretized in space. The discrete problem with this ABC has second order in time and therefore we prove necessary conditions for well posedness.…”

“…We will use the notation ABC(p, q) for the ABCs obtained when we use the Padé expansion given by a rational function p 1 (ωh)/p 2 (ωh) where p 1 and p 2 are polynomial functions with degrees p and q respectively. In this case, we define the order of absorption as the number p + q + 1 (this definition is slightly distinct to the one used in [17], where the order of absorption is defined as p + q and it corresponds to the one used in [4,5]). …”

“…The case of one dimensional Schrödinger-type equations has been studied in [4,5], when the ABCs are incorporated before the spatial discretization and in the opposite case. We remark that both two cases are very similar.…”

“…The authors show that the semidiscrete in space problems suffer from a weak ill posedness which increases when the order of absorption rises or when the spatial discretization is refined. The results in [4,5] are obtained when the order of absorption is 2 and it is numerically studied for higher order.…”

Time exponential splitting technique for the Klein–Gordon equation with Hagstrom–Warburton high-order absorbing boundary conditions

“…The ABCs are built in order to achieve, after the discretization, a stable, accurate, efficient and easy to implement scheme. There exists a wide literature on this subject, see the works [5,6,10,13,14,15,16,17] and the review papers [9,11,12,26].…”