{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T19:07:04Z","timestamp":1757617624722,"version":"3.44.0"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T00:00:00Z","timestamp":1742860800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T00:00:00Z","timestamp":1742860800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2025,8]]},"DOI":"10.1007\/s12190-025-02443-x","type":"journal-article","created":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T04:42:59Z","timestamp":1743136979000},"page":"5589-5630","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An efficient modified two-grid algorithm upon MAC scheme for simulating wormhole propagation with the Darcy-Brinkman-Forchheimer framework"],"prefix":"10.1007","volume":"71","author":[{"given":"Huishan","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1970-4951","authenticated-orcid":false,"given":"Wei","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pengshan","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,3,25]]},"reference":[{"key":"2443_CR1","doi-asserted-by":"publisher","first-page":"2171","DOI":"10.1016\/j.camwa.2017.03.002","volume":"73","author":"X Li","year":"2017","unstructured":"Li, X., Rui, H.: Characteristic block-centered finite difference method for simulating incompressible wormhole propagation. Comput. Math. Appl. 73, 2171\u20132190 (2017)","journal-title":"Comput. Math. Appl."},{"key":"2443_CR2","doi-asserted-by":"crossref","unstructured":"Sun. F., Li, X., Rui, H.: A second-order time discretizing block-centered finite difference method for compressible wormhole propagation. Numer. Meth. Part. Differ. Equ. 40, (2024)","DOI":"10.1002\/num.23091"},{"key":"2443_CR3","doi-asserted-by":"publisher","first-page":"564","DOI":"10.1016\/j.compgeo.2015.06.021","volume":"69","author":"Y Wu","year":"2015","unstructured":"Wu, Y., Salama, A., Sun, S.: Parallel simulation of wormhole propagation with the Darcy-Brinkman-Forchheimer framework. Comput. Geotech. 69, 564\u2013577 (2015)","journal-title":"Comput. Geotech."},{"key":"2443_CR4","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1017\/jfm.2019.399","volume":"872","author":"X Li","year":"2019","unstructured":"Li, X., Rui, H.: Superconvergence of a fully conservative finite difference method on non-uniform staggered grids for simulating wormhole propagation with the Darcy-Brinkman-Forchheimer framework. J. Fluid. Mech. 872, 438\u2013471 (2019)","journal-title":"J. Fluid. Mech."},{"key":"2443_CR5","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1093\/imanum\/16.3.347","volume":"16","author":"V Girault","year":"1996","unstructured":"Girault, V., Lopez, H.: Finite-element error estimates for the MAC scheme. Ima. J. Numer. Anal. 16, 347\u2013379 (1996)","journal-title":"Ima. J. Numer. Anal."},{"key":"2443_CR6","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1090\/S0025-5718-96-00665-5","volume":"65","author":"R Nicolaides","year":"1996","unstructured":"Nicolaides, R., Wu, X.: Analysis and convergence of the MAC scheme. II. Navier-Stokes. Equ. Math. Comput. 65, 29\u201344 (1996)","journal-title":"Navier-Stokes. Equ. Math. Comput."},{"key":"2443_CR7","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1137\/18M1206084","volume":"57","author":"W Chen","year":"2019","unstructured":"Chen, W., Wang, X., Yan, Y., Zhang, Z.: A second order BDF numerical scheme with variable steps for the Cahn\u2013Hilliard equation. SIAM. J. Numer. Anal. 57, 495\u2013525 (2019)","journal-title":"SIAM. J. Numer. Anal."},{"key":"2443_CR8","volume-title":"Spring Series in Computational Mathematics","author":"E Hairer","year":"1996","unstructured":"Hairer, E., Wanner, G.: Solving differential equations II: stiff and differential-algebraic problems. In::Spring Series in Computational Mathematics, 14, second, Springer, Berlin (1996)","edition":"14, second"},{"key":"2443_CR9","first-page":"1217","volume":"268","author":"Y Chen","year":"2015","unstructured":"Chen, Y., Zhou, J.: Error estimates of spectral Legendre-Galerkin methods for the fourth-order equation in one dimension. Appl. Math. Comput. 268, 1217\u20131226 (2015)","journal-title":"Appl. Math. Comput."},{"key":"2443_CR10","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1137\/18M1206084","volume":"57","author":"W Chen","year":"2019","unstructured":"Chen, W., Wang, X., Yang, Y., Zhang, Z.: A second order BDF numerical scheme with variable steps for the Cahn-Hilliard equation. SIAM. J. Numer. Anal. 57, 495\u2013525 (2019)","journal-title":"SIAM. J. Numer. Anal."},{"key":"2443_CR11","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1093\/imamat\/23.3.321","volume":"23","author":"O Axelsson","year":"1979","unstructured":"Axelsson, O., Gustafsson, I.: A modified upwind scheme for convective transport equations and the use of a conjugate gradient method for the solution of non-symmetric systems of equations. IMA. J. Appl. Math. 23, 321\u2013337 (1979)","journal-title":"IMA. J. Appl. Math."},{"key":"2443_CR12","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/j.cam.2016.10.031","volume":"317","author":"Z Zhou","year":"2017","unstructured":"Zhou, Z., Liang, D.: The mass-preserving and modified-upwind splitting DDM scheme for time-dependent convection-diffusion equations. J. Comput. Appl. Math. 317, 247\u2013273 (2017)","journal-title":"J. Comput. Appl. Math."},{"key":"2443_CR13","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1137\/0915016","volume":"15","author":"J Xu","year":"1994","unstructured":"Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM. J. Sci. Comput. 15, 231\u2013237 (1994)","journal-title":"SIAM. J. Sci. Comput."},{"key":"2443_CR14","doi-asserted-by":"publisher","first-page":"1759","DOI":"10.1137\/S0036142992232949","volume":"33","author":"J Xu","year":"1996","unstructured":"Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM. J. Numer. Anal. 33, 1759\u20131777 (1996)","journal-title":"SIAM. J. Numer. Anal."},{"key":"2443_CR15","doi-asserted-by":"publisher","first-page":"1941","DOI":"10.1137\/14097954X","volume":"53","author":"H Rui","year":"2015","unstructured":"Rui, H., Liu, W.: A two-grid block-centered finite difference method for Darcy-Forchheimer flow in porous media. SIAM. J. Numer. Anal. 53, 1941\u20131962 (2015)","journal-title":"SIAM. J. Numer. Anal."},{"key":"2443_CR16","doi-asserted-by":"publisher","first-page":"1786","DOI":"10.1007\/s10915-017-0516-6","volume":"74","author":"W Liu","year":"2018","unstructured":"Liu, W., Cui, J.: A two-grid block-centered finite difference algorithm for nonlinear compressible Darcy-Forchheimer model in porous media. J. Sci. Comput. 74, 1786\u20131815 (2018)","journal-title":"J. Sci. Comput."},{"key":"2443_CR17","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.cam.2016.10.030","volume":"315","author":"L Chen","year":"2017","unstructured":"Chen, L., Zheng, B., Lin, G., Voulgarakis, N.: A two-level stochas-tic collacation method for semilinear elliptic equations with random coefficients. J. Comput. Appl. Math. 315, 195\u2013207 (2017)","journal-title":"J. Comput. Appl. Math."},{"key":"2443_CR18","doi-asserted-by":"publisher","first-page":"409","DOI":"10.4208\/aamm.OA-2017-0011","volume":"10","author":"L Chen","year":"2018","unstructured":"Chen, L., Chen, Y.: A novel discretization method for Semilinear Reaction-Diffusion Equation. Adv. Appl. Math. Mech. 10, 409\u2013423 (2018)","journal-title":"Adv. Appl. Math. Mech."},{"key":"2443_CR19","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/s11075-019-00801-y","volume":"85","author":"W Qiu","year":"2020","unstructured":"Qiu, W., Xu, D., Guo, J., Zhou, J.: A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile\/immobile transport model. Numer. Algorithms. 85, 39\u201358 (2020)","journal-title":"Numer. Algorithms."},{"key":"2443_CR20","doi-asserted-by":"crossref","unstructured":"Cen, D., Wang, Z.: Time two-grid technique combined with temporal second order difference method for two-dimensional semilinear fractional sub-diffusion equations. Appl. Math. Lett. 129, (2022)","DOI":"10.1016\/j.aml.2022.107919"},{"key":"2443_CR21","doi-asserted-by":"publisher","first-page":"550","DOI":"10.1016\/j.matcom.2024.03.025","volume":"221","author":"Y Mei","year":"2024","unstructured":"Mei, Y., Cui, M., Zeng, F.: A time two-grid algorithm for two-dimensional nonlinear time-fractional partial integro-differential equations. Math. Comput. Simul. 221, 550\u2013569 (2024)","journal-title":"Math. Comput. Simul."},{"key":"2443_CR22","doi-asserted-by":"publisher","first-page":"1135","DOI":"10.1137\/15M1050550","volume":"55","author":"H Rui","year":"2017","unstructured":"Rui, H., Li, X.: Stability and superconvergence of MAC scheme for Stokes equations on nonuniform grids. SIAM. J. Numer. Anal. 55, 1135\u20131158 (2017)","journal-title":"SIAM. J. Numer. Anal."},{"key":"2443_CR23","doi-asserted-by":"publisher","first-page":"1100","DOI":"10.1080\/00207160.2019.1607843","volume":"97","author":"H Chen","year":"2020","unstructured":"Chen, H., Xu, D., Cao, J., Zhou, J.: A formally second order BDF ADI difference scheme for the three-dimensional time-fractional heat equation. Int. J. Comput. Math. 97, 1100\u20131117 (2020)","journal-title":"Int. J. Comput. Math."},{"key":"2443_CR24","doi-asserted-by":"crossref","unstructured":"Song Y, Liu W, Fan G: Second-order efficient algorithm for coupled nonlinear model of groundwater transport system. J. Math. Anal. Appl. 531, (2024)","DOI":"10.1016\/j.jmaa.2023.127847"},{"key":"2443_CR25","doi-asserted-by":"publisher","first-page":"202","DOI":"10.1016\/j.camwa.2024.09.024","volume":"175","author":"W Liu","year":"2024","unstructured":"Liu, W., Wang, Z., Fan, G., Song, Y.: Modified upwind finite volume scheme with second-order Lagrange multiplier method for dimensionally reduced transport model in intersecting fractured porous media. Comput. Math. Appl. 175, 202\u2013223 (2024)","journal-title":"Comput. Math. Appl."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02443-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02443-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02443-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T08:09:33Z","timestamp":1757146173000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02443-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,25]]},"references-count":25,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["2443"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02443-x","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"type":"print","value":"1598-5865"},{"type":"electronic","value":"1865-2085"}],"subject":[],"published":{"date-parts":[[2025,3,25]]},"assertion":[{"value":"2 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 February 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 February 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 March 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}