{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T06:52:59Z","timestamp":1775371979958,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T00:00:00Z","timestamp":1663804800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSF of China","award":["12061076"],"award-info":[{"award-number":["12061076"]}]},{"name":"NSF of China","award":["12126361"],"award-info":[{"award-number":["12126361"]}]},{"name":"NSF of China","award":["11701493"],"award-info":[{"award-number":["11701493"]}]},{"name":"NSF of China","award":["XJEDU2020I 001"],"award-info":[{"award-number":["XJEDU2020I 001"]}]},{"name":"NSF of China","award":["2020D04002"],"award-info":[{"award-number":["2020D04002"]}]},{"name":"Scientific Research Plan of Universities in the Autonomous Region","award":["12061076"],"award-info":[{"award-number":["12061076"]}]},{"name":"Scientific Research Plan of Universities in the Autonomous Region","award":["12126361"],"award-info":[{"award-number":["12126361"]}]},{"name":"Scientific Research Plan of Universities in the Autonomous Region","award":["11701493"],"award-info":[{"award-number":["11701493"]}]},{"name":"Scientific Research Plan of Universities in the Autonomous Region","award":["XJEDU2020I 001"],"award-info":[{"award-number":["XJEDU2020I 001"]}]},{"name":"Scientific Research Plan of Universities in the Autonomous Region","award":["2020D04002"],"award-info":[{"award-number":["2020D04002"]}]},{"name":"Key Laboratory Open Project of Xinjiang Province","award":["12061076"],"award-info":[{"award-number":["12061076"]}]},{"name":"Key Laboratory Open Project of Xinjiang Province","award":["12126361"],"award-info":[{"award-number":["12126361"]}]},{"name":"Key Laboratory Open Project of Xinjiang Province","award":["11701493"],"award-info":[{"award-number":["11701493"]}]},{"name":"Key Laboratory Open Project of Xinjiang Province","award":["XJEDU2020I 001"],"award-info":[{"award-number":["XJEDU2020I 001"]}]},{"name":"Key Laboratory Open Project of Xinjiang Province","award":["2020D04002"],"award-info":[{"award-number":["2020D04002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we consider a fully discrete modular grad-div stabilization algorithm for time-dependent thermally coupled magnetohydrodynamic (MHD) equations. The main idea of the proposed algorithm is to add an extra minimally intrusive module to penalize the divergence errors of velocity and improve the computational efficiency for increasing values of the Reynolds number and grad-div stabilization parameters. In addition, we provide the unconditional stability and optimal convergence analysis of this algorithm. Finally, several numerical experiments are performed and further indicated these advantages over the algorithm without grad-div stabilization.<\/jats:p>","DOI":"10.3390\/e24101336","type":"journal-article","created":{"date-parts":[[2022,9,22]],"date-time":"2022-09-22T21:10:05Z","timestamp":1663881005000},"page":"1336","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Modular Grad-Div Stabilization Method for Time-Dependent Thermally Coupled MHD Equations"],"prefix":"10.3390","volume":"24","author":[{"given":"Xianzhu","family":"Li","sequence":"first","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haiyan","family":"Su","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1063\/1.1457269","article-title":"Book review: An Introduction to Magnetohydrodynamics","volume":"55","author":"Davidson","year":"2002","journal-title":"Phys. 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