{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T04:59:25Z","timestamp":1771477165042,"version":"3.50.1"},"reference-count":18,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,18]],"date-time":"2020-03-18T00:00:00Z","timestamp":1584489600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["DUT19RC(4)038"],"award-info":[{"award-number":["DUT19RC(4)038"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["19K03672"],"award-info":[{"award-number":["19K03672"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2015M581689"],"award-info":[{"award-number":["2015M581689"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In addition, we introduce an extra term in the difference scheme which is used to relax the time step restriction for improving the stability condition. With the augmented terms, we prove the stability using the induction method. Numerical experiments show the validity and the accuracy of the scheme, along with its efficiency.<\/jats:p>","DOI":"10.3390\/e22030352","type":"journal-article","created":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T03:54:14Z","timestamp":1584590054000},"page":"352","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Stability Analysis of the Explicit Difference Scheme for Richards Equation"],"prefix":"10.3390","volume":"22","author":[{"given":"Fengnan","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematical and Physical Sciences, Dalian University of Technology, Panjin 124221, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7465-3168","authenticated-orcid":false,"given":"Yasuhide","family":"Fukumoto","sequence":"additional","affiliation":[{"name":"Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan"}]},{"given":"Xiaopeng","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Sciences, Northeastern University, Shenyang 110819, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"9679","DOI":"10.1002\/2017WR021097","article-title":"Exact Solutions of the Richards Equation With Nonlinear Plant-Root Extraction","volume":"53","author":"Broadbridge","year":"2017","journal-title":"Water Resour. 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