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

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.

33 pages, 10 figures

References in corpus (12)

• The Hele-Shaw asymptotics for mechanical models of tumor growth
• Finite element systems of differential forms
• An Energetic Variational Approach for the Cahn--Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis
• An Energetic Variational Approach for ion transport
• A Study on Moving Mesh Finite Element Solution of the Porous Medium Equation
• Numerical Study of a Particle Method for Gradient Flows
• Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
• A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes
• Diffusion limit of kinetic equations for multiple species charged particles
• Variational Particle Schemes for the Porous Medium Equation and for the System of Isentropic Euler Equations
• Numerical methods for Porous Medium Equation by an Energetic Variational Approach
• Adaptive Finite Element Solution of the Porous Medium Equation in Pressure Formulation

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