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In this paper, we present a systematic framework to derive a Lagrangianscheme for porous medium type generalized diffusion equations by employing adiscrete energetic variational approach. Such discrete energetic variationalapproaches are analogous to energetic variational approaches in a semidiscretelevel, which provide a basis of deriving the "semi-discrete equations" and canbe applied to a large class of partial differential equations withenergy-dissipation laws and kinematic relations. The numerical schemes derivedby this framework can inherit various properties from the continuousenergy-dissipation law, such as conservation of mass and the dissipation of thediscrete energy. As an illustration, we develop two numerical schemes for themultidimensional porous medium equations (PME), based on two differentenergy-dissipation laws. We focus on the numerical scheme based on theenergy-dissipation law with $\frac{1}{2} \int_{\Omega} |\mathbf{u}|^2\mathrm{d} \mathbf{x}$ as the dissipation. Several numerical experimentsdemonstrate the accuracy of this scheme as well as its ability in capturing thefree boundary and estimating the waiting time for the PME in both 1D and 2D.

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