数学
方案(数学)
有限元法
非线性系统
最大值原理
应用数学
数学分析
数学优化
最优控制
结构工程
物理
量子力学
工程类
作者
István Faragó,Róbert Horváth,János Karátson
出处
期刊:Ima Journal of Numerical Analysis
日期:2024-09-27
标识
DOI:10.1093/imanum/drae072
摘要
Abstract In this paper, we extend our earlier results in Faragó, I., Karátson, J. and Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems. IMA J. Numer. Anal., 32, 1541–1573) on the discrete maximum-minimum principle (DMP) for nonlinear parabolic systems of PDEs. We propose a linearly implicit scheme, where only linear problems have to be solved on the time layers. We obtain a DMP without the restrictive condition $\varDelta t\le O(h^{2})$. We show that we only need the lower bound $\varDelta t\ge O(h^{2})$, further, depending on the Lipschitz condition of the given nonlinearity, the upper bound is just $\varDelta t\le C$ (for globally Lipschitz) or $\varDelta t\le O(h^{\gamma })$ (for locally Lipschitz) for some constant $C>0$ arising from the PDE, or some $\gamma < 2$, respectively. In most situations in practical models, the latter condition becomes $\varDelta t \le O( h^{2/3} )$ in 2D and $\varDelta t \le O( h )$ in 3D. Various real-life examples are also presented where the results can be applied to obtain physically relevant numerical solutions.
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