{"title":"拟线性次扩散方程的线性正则化有限差分格式","authors":"A. Lapin, E. Laitinen","doi":"10.1515/rnam-2022-0019","DOIUrl":null,"url":null,"abstract":"Abstract A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with a time-fractional derivative and coefficients at the elliptic part that depend on the gradient of the solution is considered. Conditions on the coefficients ensure the monotonicity and Lipschitz property of the elliptic operator on the set of functions whose gradients in space variables are uniformly bounded. For this problem, a linear regularized mesh scheme is constructed and investigated. A sufficient condition is derived for the regularization parameter that ensures the so-called local correctness of the mesh scheme. On the basis of correctness and approximation estimates for model problems with time-fractional Caputo or Caputo–Fabrizio derivatives, accuracy estimates are given in terms of mesh and regularization parameters under the assumption of the existence of a smooth solution to the differential problem. The presented results of the numerical experiments confirm the obtained asymptotic accuracy estimates.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"221 - 229"},"PeriodicalIF":0.5000,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Linear regularized finite difference scheme for the quasilinear subdiffusion equation\",\"authors\":\"A. Lapin, E. Laitinen\",\"doi\":\"10.1515/rnam-2022-0019\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with a time-fractional derivative and coefficients at the elliptic part that depend on the gradient of the solution is considered. Conditions on the coefficients ensure the monotonicity and Lipschitz property of the elliptic operator on the set of functions whose gradients in space variables are uniformly bounded. For this problem, a linear regularized mesh scheme is constructed and investigated. A sufficient condition is derived for the regularization parameter that ensures the so-called local correctness of the mesh scheme. On the basis of correctness and approximation estimates for model problems with time-fractional Caputo or Caputo–Fabrizio derivatives, accuracy estimates are given in terms of mesh and regularization parameters under the assumption of the existence of a smooth solution to the differential problem. The presented results of the numerical experiments confirm the obtained asymptotic accuracy estimates.\",\"PeriodicalId\":49585,\"journal\":{\"name\":\"Russian Journal of Numerical Analysis and Mathematical Modelling\",\"volume\":\"37 1\",\"pages\":\"221 - 229\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Russian Journal of Numerical Analysis and Mathematical Modelling\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/rnam-2022-0019\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Numerical Analysis and Mathematical Modelling","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/rnam-2022-0019","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Linear regularized finite difference scheme for the quasilinear subdiffusion equation
Abstract A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with a time-fractional derivative and coefficients at the elliptic part that depend on the gradient of the solution is considered. Conditions on the coefficients ensure the monotonicity and Lipschitz property of the elliptic operator on the set of functions whose gradients in space variables are uniformly bounded. For this problem, a linear regularized mesh scheme is constructed and investigated. A sufficient condition is derived for the regularization parameter that ensures the so-called local correctness of the mesh scheme. On the basis of correctness and approximation estimates for model problems with time-fractional Caputo or Caputo–Fabrizio derivatives, accuracy estimates are given in terms of mesh and regularization parameters under the assumption of the existence of a smooth solution to the differential problem. The presented results of the numerical experiments confirm the obtained asymptotic accuracy estimates.
期刊介绍:
The Russian Journal of Numerical Analysis and Mathematical Modelling, published bimonthly, provides English translations of selected new original Russian papers on the theoretical aspects of numerical analysis and the application of mathematical methods to simulation and modelling. The editorial board, consisting of the most prominent Russian scientists in numerical analysis and mathematical modelling, selects papers on the basis of their high scientific standard, innovative approach and topical interest.
Topics:
-numerical analysis-
numerical linear algebra-
finite element methods for PDEs-
iterative methods-
Monte-Carlo methods-
mathematical modelling and numerical simulation in geophysical hydrodynamics, immunology and medicine, fluid mechanics and electrodynamics, geosciences.