{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T23:11:13Z","timestamp":1771024273477,"version":"3.50.1"},"reference-count":39,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,30]],"date-time":"2024-08-30T00:00:00Z","timestamp":1724976000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"College Students Innovations Special Project funded by Northeast Forestry University of China","award":["DC-2023166"],"award-info":[{"award-number":["DC-2023166"]}]},{"name":"College Students Innovations Special Project funded by Northeast Forestry University of China","award":["LH2022A002"],"award-info":[{"award-number":["LH2022A002"]}]},{"name":"Natural Science Foundation of Heilongjiang Province of China","award":["DC-2023166"],"award-info":[{"award-number":["DC-2023166"]}]},{"name":"Natural Science Foundation of Heilongjiang Province of China","award":["LH2022A002"],"award-info":[{"award-number":["LH2022A002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we present a novel explicit structure-preserving numerical method for solving nonlinear space-fractional Schr\u00f6dinger equations based on the concept of the scalar auxiliary variable approach. Firstly, we convert the equations into an equivalent system through the introduction of a scalar variable. Subsequently, a semi-discrete energy-preserving scheme is developed by employing a fourth-order fractional difference operator to discretize the equivalent system in spatial direction, and obtain the fully discrete version by using an explicit relaxed Runge\u2013Kutta method for temporal integration. The proposed method preserves the energy conservation property of the space-fractional nonlinear Schr\u00f6dinger equation and achieves high accuracy. Numerical experiments are carried out to verify the structure-preserving qualities of the proposed method.<\/jats:p>","DOI":"10.3390\/axioms13090591","type":"journal-article","created":{"date-parts":[[2024,8,30]],"date-time":"2024-08-30T07:45:47Z","timestamp":1725003947000},"page":"591","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Energy-Conserving Explicit Relaxed Runge\u2013Kutta Methods for the Fractional Nonlinear Schr\u00f6dinger Equation Based on Scalar Auxiliary Variable Approach"],"prefix":"10.3390","volume":"13","author":[{"given":"Yizhuo","family":"Zhao","sequence":"first","affiliation":[{"name":"Department of Mathematics, Northeast Forestry University, Harbin 150040, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4993-7377","authenticated-orcid":false,"given":"Yu","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Northeast Forestry University, Harbin 150040, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-4917-9122","authenticated-orcid":false,"given":"Jiaxin","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Northeast Forestry University, Harbin 150040, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yang","family":"Cao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Northeast Forestry University, Harbin 150040, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"110201","DOI":"10.1088\/1402-4896\/acfe73","article-title":"Advanced fractional calculus, differential equations and neural networks: Analysis, modeling and numerical computations","volume":"98","author":"Baleanu","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"082104","DOI":"10.1063\/1.2235026","article-title":"Some physical applications of fractional Schr\u00f6dinger equation","volume":"47","author":"Guo","year":"2006","journal-title":"J. Math. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"100805","DOI":"10.1016\/j.padiff.2024.100805","article-title":"Multicomponent nonlinear fractional Schr\u00f6dinger equation: On the study of optical wave propagation in the fiber optics","volume":"11","author":"Muhammad","year":"2024","journal-title":"Partial. Differ. Equ. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.camwa.2023.04.009","article-title":"Two novel conservative exponential relaxation methods for the space-fractional nonlinear Schr\u00f6dinger equation","volume":"142","author":"Xu","year":"2023","journal-title":"Comput. Math. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2257","DOI":"10.1016\/j.camwa.2015.12.042","article-title":"Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schr\u00f6dinger equation","volume":"71","author":"Duo","year":"2016","journal-title":"Comput. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1016\/j.apnum.2019.11.008","article-title":"Energy preserving relaxation method for space-fractional nonlinear Schr\u00f6dinger equation","volume":"152","author":"Cheng","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"109009","DOI":"10.1016\/j.jcp.2019.109009","article-title":"What is the fractional Laplacian? A comparative review with new results","volume":"404","author":"Lischke","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/S0377-0427(96)00156-2","article-title":"A finite-difference method for the numerical solution of the Schr\u00f6dinger equation","volume":"79","author":"Simos","year":"1997","journal-title":"J. Comput. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1137\/110830800","article-title":"Uniform error estimates of finite difference methods for the nonlinear Schr\u00f6dinger equation with wave operator","volume":"50","author":"Bao","year":"2012","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1779","DOI":"10.1137\/S0036142997330111","article-title":"A space-time finite element method for the nonlinear Schr\u00f6dinger equation: The continuous Galerkin method","volume":"36","author":"Karakashian","year":"1999","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_11","first-page":"125141","article-title":"Nonconforming finite element method for a generalized nonlinear Schr\u00f6dinger equation","volume":"377","author":"Zhang","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3231","DOI":"10.1137\/120866373","article-title":"Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schr\u00f6dinger equations","volume":"50","author":"Thalhammer","year":"2012","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.jcp.2017.09.028","article-title":"A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives","volume":"351","year":"2017","journal-title":"J. Comput. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.apnum.2018.03.008","article-title":"Structure-preserving numerical methods for the fractional Schr\u00f6dinger equation","volume":"129","author":"Wang","year":"2018","journal-title":"Appl. Numer. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.cnsns.2017.10.019","article-title":"An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions","volume":"59","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"A1026","DOI":"10.1137\/22M152178X","article-title":"High-Order mass-and energy-conserving methods for the nonlinear Schr\u00f6dinger equation","volume":"46","author":"Bai","year":"2024","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.camwa.2021.06.018","article-title":"On convergence of a structure preserving difference scheme for two-dimensional space-fractional nonlinear Schr\u00f6dinger equation and its fast implementation","volume":"98","author":"Hu","year":"2021","journal-title":"Comput. Math. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1444","DOI":"10.1007\/s10915-018-0708-8","article-title":"On the conservation of fractional nonlinear Schr\u00f6dinger equation\u2019s invariants by the local discontinuous Galerkin method","volume":"77","author":"Castillo","year":"2018","journal-title":"J. Sci. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.apnum.2016.09.009","article-title":"An improved collocation method for multi-dimensional space\u2013time variable-order fractional Schr\u00f6dinger equations","volume":"111","author":"Bhrawy","year":"2017","journal-title":"Appl. Numer. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1007\/s10915-019-01050-w","article-title":"Error analysis and numerical simulations of Strang splitting method for space fractional nonlinear Schr\u00f6dinger equation","volume":"81","author":"Zhai","year":"2019","journal-title":"J. Sci. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"644","DOI":"10.1016\/j.jcp.2014.04.047","article-title":"A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schr\u00f6dinger equations","volume":"272","author":"Wang","year":"2014","journal-title":"J. Comput. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.matcom.2023.09.001","article-title":"Arbitrarily high-order explicit energy-conserving methods for the generalized nonlinear fractional Schr\u00f6dinger wave equations","volume":"216","author":"Liu","year":"2024","journal-title":"Math. Comput. Simul."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"934","DOI":"10.1137\/S0036142901396521","article-title":"A relaxation scheme for the nonlinear Schr\u00f6dinger equation","volume":"42","author":"Besse","year":"2004","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_24","first-page":"817","article-title":"Featured review: Geometric numerical integration: Structure-preserving algorithms for ordinary differential equations","volume":"45","author":"McLachlan","year":"2003","journal-title":"SIAM Rev."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Leimkuhler, B., and Reich, S. (2004). Simulating Hamiltonian Dynamics, Cambridge University Press. Number 14.","DOI":"10.1017\/CBO9780511614118"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"111925","DOI":"10.1016\/j.jcp.2023.111925","article-title":"Linearly implicit and high-order energy-preserving relaxation schemes for highly oscillatory Hamiltonian systems","volume":"477","author":"Li","year":"2023","journal-title":"J. Comput. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.matcom.2020.12.025","article-title":"High-order structure-preserving algorithms for the multi-dimensional fractional nonlinear Schr\u00f6dinger equation based on the SAV approach","volume":"185","author":"Fu","year":"2021","journal-title":"Math. Comput. Simul."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.apnum.2014.11.007","article-title":"A weighted finite difference method for the fractional diffusion equation based on the Riemann\u2013Liouville derivative","volume":"90","author":"Sousa","year":"2015","journal-title":"Appl. Numer. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1418","DOI":"10.1137\/130933447","article-title":"Fourth order accurate scheme for the space fractional diffusion equations","volume":"52","author":"Chen","year":"2014","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1137\/0517050","article-title":"Discretized fractional calculus","volume":"17","author":"Lubich","year":"1986","journal-title":"SIAM J. Math. Anal."},{"key":"ref_31","first-page":"124","article-title":"An explicit fourth-order energy-preserving scheme for Riesz space fractional nonlinear wave equations","volume":"351","author":"Zhao","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Li, Y., Shan, W., and Zhang, Y. (2022). High-Order Dissipation-Preserving Methods for Nonlinear Fractional Generalized Wave Equations. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6050264"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2850","DOI":"10.1137\/19M1263662","article-title":"Relaxation Runge\u2013Kutta methods: Conservation and stability for inner-product norms","volume":"57","author":"Ketcheson","year":"2019","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"A612","DOI":"10.1137\/19M1263480","article-title":"Relaxation Runge\u2013Kutta methods: Fully discrete explicit entropy-stable schemes for the compressible Euler and Navier\u2013Stokes equations","volume":"42","author":"Ranocha","year":"2020","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"A2865","DOI":"10.1137\/140961560","article-title":"A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrodinger equation","volume":"36","author":"Zhao","year":"2014","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4462","DOI":"10.1002\/mma.4317","article-title":"Numerical approximation of the conservative Allen\u2013Cahn equation by operator splitting method","volume":"40","author":"Weng","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_37","unstructured":"Willoughby, M.R. (2011). High-Order Time-Adaptive Numerical Methods for the Allen-Cahn and Cahn-Hilliard Equations. [Ph.D. Thesis, University of British Columbia]."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1114","DOI":"10.1016\/j.camwa.2016.01.022","article-title":"Split-step alternating direction implicit difference scheme for the fractional Schr\u00f6dinger equation in two dimensions","volume":"71","author":"Wang","year":"2016","journal-title":"Comput. Math. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"4438","DOI":"10.1016\/j.camwa.2018.03.042","article-title":"An efficient Fourier spectral exponential time differencing method for the space-fractional nonlinear Schr\u00f6dinger equations","volume":"75","author":"Liang","year":"2018","journal-title":"Comput. Math. Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/591\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:45:33Z","timestamp":1760111133000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/591"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,30]]},"references-count":39,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["axioms13090591"],"URL":"https:\/\/doi.org\/10.3390\/axioms13090591","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,30]]}}}