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Volume 33, Issue 4
Efficient Splitting Methods Based on Modified Potentials: Numerical Integration of Linear Parabolic Problems and Imaginary Time Propagation of the Schrödinger Equation

Sergio Blanes, Fernando Casas, Cesáreo González & Mechthild Thalhammer

DOI: 10.4208/cicp.OA-2022-0247

Commun. Comput. Phys., 33 (2023), pp. 937-961.

Published online: 2023-05

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  • Abstract

We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and, in particular, for the time dependent Schrödinger equation, both in real and imaginary time. They are based on the use of a double commutator and a modified processor, and are more efficient than other widely used schemes found in the literature. Moreover, for certain potentials, they achieve order six. Several examples in one, two and three dimensions clearly illustrate the computational advantages of the new schemes.

  • Keywords

Schrödinger equation, imaginary time propagation, parabolic equations, operator splitting methods, modified potentials.

  • AMS Subject Headings

65L05, 65M12, 65J10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-33-937, author = {Blanes , SergioCasas , FernandoGonzález , Cesáreo and Thalhammer , Mechthild}, title = {Efficient Splitting Methods Based on Modified Potentials: Numerical Integration of Linear Parabolic Problems and Imaginary Time Propagation of the Schrödinger Equation}, journal = {Communications in Computational Physics}, year = {2023}, volume = {33}, number = {4}, pages = {937--961}, abstract = {

We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and, in particular, for the time dependent Schrödinger equation, both in real and imaginary time. They are based on the use of a double commutator and a modified processor, and are more efficient than other widely used schemes found in the literature. Moreover, for certain potentials, they achieve order six. Several examples in one, two and three dimensions clearly illustrate the computational advantages of the new schemes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0247}, url = {http://global-sci.org/intro/article_detail/cicp/21665.html} }
TY - JOUR T1 - Efficient Splitting Methods Based on Modified Potentials: Numerical Integration of Linear Parabolic Problems and Imaginary Time Propagation of the Schrödinger Equation AU - Blanes , Sergio AU - Casas , Fernando AU - González , Cesáreo AU - Thalhammer , Mechthild JO - Communications in Computational Physics VL - 4 SP - 937 EP - 961 PY - 2023 DA - 2023/05 SN - 33 DO - http://doi.org/10.4208/cicp.OA-2022-0247 UR - https://global-sci.org/intro/article_detail/cicp/21665.html KW - Schrödinger equation, imaginary time propagation, parabolic equations, operator splitting methods, modified potentials. AB -

We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and, in particular, for the time dependent Schrödinger equation, both in real and imaginary time. They are based on the use of a double commutator and a modified processor, and are more efficient than other widely used schemes found in the literature. Moreover, for certain potentials, they achieve order six. Several examples in one, two and three dimensions clearly illustrate the computational advantages of the new schemes.

Blanes , SergioCasas , FernandoGonzález , Cesáreo and Thalhammer , Mechthild. (2023). Efficient Splitting Methods Based on Modified Potentials: Numerical Integration of Linear Parabolic Problems and Imaginary Time Propagation of the Schrödinger Equation. Communications in Computational Physics. 33 (4). 937-961. doi:10.4208/cicp.OA-2022-0247
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