{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T14:09:33Z","timestamp":1782310173979,"version":"3.54.5"},"reference-count":49,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,2]],"date-time":"2025-07-02T00:00:00Z","timestamp":1751414400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["U2242214"],"award-info":[{"award-number":["U2242214"]}]},{"name":"National Natural Science Foundation of China","award":["K3-949"],"award-info":[{"award-number":["K3-949"]}]},{"name":"Fujian Provincial Units Special Funds for Education and Research","award":["U2242214"],"award-info":[{"award-number":["U2242214"]}]},{"name":"Fujian Provincial Units Special Funds for Education and Research","award":["K3-949"],"award-info":[{"award-number":["K3-949"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A mesoscopic lattice Boltzmann method based on the BGK model is proposed to solve a class of two-dimensional second-order nonlinear partial differential equations by incorporating an amending function. The model provides an efficient and stable framework for simulating initial value problems of second-order nonlinear partial differential equations and is adaptable to various nonlinear systems, including strongly nonlinear cases. The numerical characteristics and evolution patterns of these nonlinear equations are systematically investigated. A D2Q4 lattice model is employed, and the kinetic moment constraints for both local equilibrium and correction distribution functions are derived in the four velocity directions. Explicit analytical expressions for these distribution functions are presented. The model is verified to recover the target macroscopic equations in the continuous limit via Chapman\u2013Enskog analysis. Numerical experiments using exact solutions are performed to assess the model\u2019s accuracy and stability. The results show excellent agreement with exact solutions and demonstrate the model\u2019s robustness in capturing nonlinear dynamics.<\/jats:p>","DOI":"10.3390\/e27070717","type":"journal-article","created":{"date-parts":[[2025,7,2]],"date-time":"2025-07-02T06:10:26Z","timestamp":1751436626000},"page":"717","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Lattice Boltzmann BGK Model with an Amending Function for Two-Dimensional Second-Order Nonlinear Partial Differential Equations"],"prefix":"10.3390","volume":"27","author":[{"given":"Xiaohua","family":"Bi","sequence":"first","affiliation":[{"name":"School of Liberal Arts and Sciences, North China Institute of Aerospace Engineering, Langfang 065000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Junbo","family":"Lei","sequence":"additional","affiliation":[{"name":"Fujian Key Laboratory of Analytical Mathematics and Applications (FJKLAMA), Center for Applied Mathematics of Fujian Province (FJNU), Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Demei","family":"Li","sequence":"additional","affiliation":[{"name":"Fujian Key Laboratory of Analytical Mathematics and Applications (FJKLAMA), Center for Applied Mathematics of Fujian Province (FJNU), Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lindong","family":"Lai","sequence":"additional","affiliation":[{"name":"Center of Philippine Studies, Fujian Normal University, Fuzhou 350117, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5978-5736","authenticated-orcid":false,"given":"Huilin","family":"Lai","sequence":"additional","affiliation":[{"name":"Fujian Key Laboratory of Analytical Mathematics and Applications (FJKLAMA), Center for Applied Mathematics of Fujian Province (FJNU), Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhipeng","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Science, Tianjin Chengjian University, Tianjin 300384, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Debnath, L. 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