Arnold, AntonEhrhardt, MatthiasSofronov, Ivan2022-05-112022-05-112002-12-062197-8085https://depositonce.tu-berlin.de/handle/11303/16903http://dx.doi.org/10.14279/depositonce-15681This paper is concerned with transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation in one and two dimensions. Discrete TBCs are introduced in the numerical simulations of whole space problems in order to reduce the computational domain to a finite region. Since the discrete TBC for the Schrödinger equation includes a convolution w.r.t. time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate TBCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove stability of the resulting initial-boundary value scheme, give error estimates for the considered approximation of the boundary condition, and illustrate the efficiency of the proposed method on several examples.en510 MathematikSchrödinger equationtransparent boundary conditionsdiscrete convolutionsum of exponentialsPadé approximationsfinite difference schemesResearch Paper