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.

中文翻译：

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弱时间限制下非线性抛物型有限元问题线性隐式格式的离散最大最小原理


在本文中，我们扩展了 Faragó, I.、Karátson, J. 和 Korotov, S. 的早期结果（2012，非线性抛物型 PDE 系统的离散最大原理。IMA J. Numer. Anal., 32, 1541–1573）偏微分方程非线性抛物线系统的离散最大最小原理（DMP）。我们提出了一种线性隐式方案，其中只需要在时间层上解决线性问题。我们得到一个没有限制条件 $\varDelta t\le O(h^{2})$ 的 DMP。我们证明我们只需要下界 $\varDelta t\ge O(h^{2})$，此外，根据给定非线性的 Lipschitz 条件，上限只是 $\varDelta t\le C$ （对于全局 Lipschitz）或 $\varDelta t\le O(h^{\gamma })$ （对于局部 Lipschitz）对于偏微分方程产生的某个常数 $C>0$，或某个 $\gamma < 2$，分别。在实际模型的大多数情况下，后一个条件在 2D 中变为 $\varDelta t \le O( h^{2/3} )$，在 3D 中变为 $\varDelta t \le O( h )$。还提供了各种现实生活中的示例，其中的结果可用于获得物理相关的数值解。