On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach
On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach
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DOI：
10.1016/j.jcp.2020.109566
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发表时间：
2019-05
期刊：
影响因子：
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通讯作者：
Chun Liu;Yiwei Wang
中科院分区：
文献类型：
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作者：
Chun Liu;Yiwei Wang
本文利用离散能量变分方法，给出了求解多孔介质型广义扩散方程的变分拉格朗日格式的系统框架。这种离散能量变分方法类似于半离散水平上的能量变分方法[39]，[25]，其提供了导出变分“半离散方程”的基础，并且可以应用于具有能量变分结构的一大类偏微分方程。由此框架导出的数值格式可以继承连续能量耗散律的变分结构。作为一个例子，我们开发了两个变分拉格朗日格式的多维多孔介质方程（PME），基于两个不同的能量耗散定律。重点研究了基于能量耗散律的数值格式，|u| 2 d x作为耗散函数。几个数值实验表明，该计划的准确性，以及它的能力，在捕捉自由边界和估计的等待时间的PME在1D和2D。
In this paper, we present a systematic framework to derive a variational Lagrangian scheme for porous medium type generalized diffusion equations by employing a discrete energetic variational approach. Such discrete energetic variational approaches are analogous to energetic variational approaches [39],[25] in a semidiscrete level, which provide a basis of deriving variational “semi-discrete equations” and can be applied to a large class of partial differential equations with energetic variational structures. The numerical schemes derived by this framework can inherit the variational structure from the continuous energy-dissipation law. As an illustration, we develop two variational Lagrangian schemes for the multidimensional porous medium equations (PME), based on two different energy-dissipation laws. We focus on the numerical scheme based on the energy-dissipation law with 1 2∫ Ω| u| 2 d x as the dissipation functional. Several numerical experiments demonstrate the accuracy of this scheme as well as its ability in capturing the free boundary and estimating the waiting time for the PME in both 1D and 2D.