{"title":"A Structure-Preserving JKO Scheme for the SizeModified Poisson-Nernst-Planck-Cahn-Hilliard Equations","authors":"Jie Ding null, Xiang Ji","doi":"10.4208/nmtma.oa-2022-0078","DOIUrl":null,"url":null,"abstract":". In this paper, we propose a structure-preserving numerical scheme for the size-modified Poisson-Nernst-Planck-Cahn-Hilliard (SPNPCH) equations derived from the free energy including electrostatic energies, entropies, steric energies, and Cahn-Hilliard mixtures. Based on the Jordan-Kinderlehrer-Otto (JKO) framework and the Benamou-Brenier formula of quadratic Wasserstein distance, the SPNPCH equations are transformed into a constrained optimization problem. By exploiting the convexity of the objective function, we can prove the existence and uniqueness of the numerical solution to the optimization problem. Mass conservation and unconditional energy-dissipation are preserved automatically by this scheme. Furthermore, by making use of the singularity of the entropy term which keeps the concentration from approaching zero, we can ensure the positivity of concentration. To solve the optimization problem, we apply the quasi-Newton method, which can ensure the positivity of concentration in the iterative process. Numerical tests are performed to confirm the anticipated accuracy and the desired physical properties of the developed scheme. Finally, the proposed scheme can also be applied to study the influence of ionic sizes and gradient energy coefficients on ion distribution.","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Mathematics-Theory Methods and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4208/nmtma.oa-2022-0078","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
. In this paper, we propose a structure-preserving numerical scheme for the size-modified Poisson-Nernst-Planck-Cahn-Hilliard (SPNPCH) equations derived from the free energy including electrostatic energies, entropies, steric energies, and Cahn-Hilliard mixtures. Based on the Jordan-Kinderlehrer-Otto (JKO) framework and the Benamou-Brenier formula of quadratic Wasserstein distance, the SPNPCH equations are transformed into a constrained optimization problem. By exploiting the convexity of the objective function, we can prove the existence and uniqueness of the numerical solution to the optimization problem. Mass conservation and unconditional energy-dissipation are preserved automatically by this scheme. Furthermore, by making use of the singularity of the entropy term which keeps the concentration from approaching zero, we can ensure the positivity of concentration. To solve the optimization problem, we apply the quasi-Newton method, which can ensure the positivity of concentration in the iterative process. Numerical tests are performed to confirm the anticipated accuracy and the desired physical properties of the developed scheme. Finally, the proposed scheme can also be applied to study the influence of ionic sizes and gradient energy coefficients on ion distribution.
期刊介绍:
Numerical Mathematics: Theory, Methods and Applications (NM-TMA) publishes high-quality original research papers on the construction, analysis and application of numerical methods for solving scientific and engineering problems. Important research and expository papers devoted to the numerical solution of mathematical equations arising in all areas of science and technology are expected. The journal originates from the journal Numerical Mathematics: A Journal of Chinese Universities (English Edition). NM-TMA is a refereed international journal sponsored by Nanjing University and the Ministry of Education of China. As an international journal, NM-TMA is published in a timely fashion in printed and electronic forms.