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NUMERICAL COMPARISON OF MASS-CONSERVATIVE SCHEMES FOR THE GROSS-PITAEVSKII EQUATION
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0002-6432-5504
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0003-2598-6809
2019 (English)In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 12, no 6, p. 1247-1271Article in journal (Refereed) Published
Abstract [en]

In this paper we present a numerical comparison of various mass-conservative discretizations for the time-dependent Gross-Pitaevskii equation. We have three main objectives. First, we want to clarify how purely mass-conservative methods perform compared to methods that are additionally energy-conservative or symplectic. Second, we shall compare the accuracy of energy-conservative and symplectic methods among each other. Third, we will investigate if a linearized energy-conserving method suffers from a loss of accuracy compared to an approach which requires to solve a full nonlinear problem in each time-step. In order to obtain a representative comparison, our numerical experiments cover different physically relevant test cases, such as traveling solitons, stationary multi-solitons, Bose-Einstein condensates in an optical lattice and vortex pattern in a rapidly rotating superfluid. We shall also consider a computationally severe test case involving a pseudo Mott insulator. Our space discretization is based on finite elements throughout the paper. We will also give special attention to long time behavior and possible coupling conditions between time-step sizes and mesh sizes. The main observation of this paper is that mass conservation alone will not lead to a competitive method in complex settings. Furthermore, energy-conserving and symplectic methods are both reliable and accurate, yet, the energy-conservative schemes achieve a visibly higher accuracy in our test cases. Finally, the scheme that performs best throughout our experiments is an energy-conserving relaxation scheme with linear time-stepping proposed by C. Besse (SINUM,42(3):934-952,2004).

Place, publisher, year, edition, pages
AMER INST MATHEMATICAL SCIENCES-AIMS , 2019. Vol. 12, no 6, p. 1247-1271
Keywords [en]
Conservative schemes, Gross-Pitaevskii equation, nonlinear Schrodinger equation, numerical comparison, finite elements
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-261286DOI: 10.3934/krm.2019048ISI: 000486331900003OAI: oai:DiVA.org:kth-261286DiVA, id: diva2:1359431
Note

QC 20191009

Available from: 2019-10-09 Created: 2019-10-09 Last updated: 2019-10-09Bibliographically approved

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Henning, PatrickWärnegård, Johan

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