On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach

JOURNAL OF COMPUTATIONAL PHYSICS

卷:417

Article

Liu, Chun¹  Wang, Yiwei¹ 

[1] IIT, Dept Appl Math, Chicago, IL 60616 USA

关键词: Energetic variational approach;    Porous medium type diffusion;    Variational Lagrangian scheme;    Finite speed propagation of free boundary;    Waiting time;   

DOI  :  10.1016/j.jcp.2020.109566

来源: Elsevier

【 摘 要 】

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 energydissipation law. As an illustration, we develop two variational Lagrangian schemes for the multidimensional porous medium equations (PME), based on two different energydissipation laws. We focus on the numerical scheme based on the energy-dissipation law with 1/2 integral(Omega) vertical bar u vertical bar(2)dx 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. (c) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free