On Lagrangian schemes for the multidimensional porous medium equations by a discrete energetic variational approach
Chun Liu, Yiwei Wang · arXiv (Cornell University) · 2019
In this paper, we present a systematic framework to derive a Lagrangian scheme for general diffusion in multiple spatial dimensions by employing a discrete energetic variational approach. Such discrete energetic variational approaches are analogous to energetic variational approaches in a semidiscrete level, which provide a basis of deriving the semi-discrete equations and can be applied to a large class of partial differential with energy-dissipation laws and kinematic relations. The numerical schemes derived by this framework can inherit various properties from the continuous energy-dissipation law, such as conservation of mass and the dissipation of the discrete energy. As an illustration, we develop two numerical schemes for the multidimensional porous medium (PME), based on two different energy-dissipation laws. We focus on the numerical scheme based on the energy-dissipation law with $\frac{1}{2} \int_{\Omega} |\mathbf{u}|^2 \mathrm{d} \mathbf{x}$ as the dissipation. 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.