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

## TL;DR

This paper introduces a systematic Lagrangian scheme for porous medium equations using a discrete energetic variational approach, ensuring properties like energy dissipation and mass conservation, and demonstrates its effectiveness through numerical experiments.

## Contribution

It develops a novel framework for deriving energy-dissipative Lagrangian schemes for generalized diffusion equations, applicable to multidimensional porous medium equations.

## Key findings

- The scheme accurately captures free boundaries in PME.
- It effectively estimates waiting times in simulations.
- Numerical results confirm the scheme's stability and energy dissipation properties.

## Abstract

In this paper, we present a systematic framework to derive a 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 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 equations 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 equations (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.