Numerical approximation of the Schrodinger equation with concentrated potential

JOURNAL OF COMPUTATIONAL PHYSICS

卷:405

Article

Banjai, L.¹  Lopez-Fernandez, M.^2,3 

[1] Heriot Watt Univ, Sch Math & Comp Sci, Maxwell Inst Math Sci, Edinburgh EH14 4AS, Midlothian, Scotland

[2] Univ Malaga, Fac Sci, Dept Math Anal Stat & OR & Appl Math, Malaga, Spain

[3] Sapienza Univ Rome, Dept Math Guido Castelnuovo, Rome, Italy

关键词: Fast and oblivious algorithms;    Convolution quadrature;    Schrodinger equation;    Boundary integral equations;    Contour integral methods;   

DOI  :  10.1016/j.jcp.2019.109155

来源: Elsevier

【 摘 要 】

We present a family of algorithms for the numerical approximation of the Schrodinger equation with potential concentrated at a finite set of points. Our methods belong to the so-called fast and oblivious convolution quadrature algorithms. These algorithms are special implementations of Lubich's Convolution Quadrature which allow, for certain applications in particular parabolic problems, to significantly reduce the computational cost and memory requirements. Recently it has been noticed that their use can be extended to some hyperbolic problems. Here we propose a new family of such efficient algorithms tailored to the features of the Green's function for Schrodinger equations. In this way, we are able to keep the computational cost and the storage requirements significantly below existing approaches. These features allow us to perform reliable numerical simulations for longer times even in cases where the solution becomes highly oscillatory or seems to develop finite time blow-up. We illustrate our new algorithm with several numerical experiments. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free