Paper Info
Title
Numerical methods for Porous Medium Equation by an Energetic Variational Approach
Abstract
We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem difficult to handle. Based on different dissipative energy laws, we develop two numerical schemes by an energetic variational approach. Firstly, based on flog⁡f as the total energy form of the dissipative law, we obtain the trajectory equation, and then construct a fully discrete scheme. It is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy. Next, based on 1/(2f) as the total energy form of the dissipation law, we construct a linear numerical scheme for the corresponding trajectory equation. Both schemes preserve the corresponding discrete dissipation law. Meanwhile, under some smoothness assumption, both schemes are second-order convergent in space and first-order convergent in time. Each scheme yields a good approximation for the solution and the free boundary. No oscillation is observed for the numerical solution around the free boundary. Furthermore, the waiting time problem could be naturally treated, which has been a well-known difficult issue for all the existing methods. Due to its linear nature, the second scheme is more efficient.
Year
DOI
Venue
2019
10.1016/j.jcp.2019.01.055
Journal of Computational Physics
Keywords
Field
DocType
Energetic variational approach,Porous medium equation,Finite speed propagation of free boundary,Waiting time,Trajectory equation
Oscillation,Trajectory of a projectile,Mathematical analysis,Dissipation,Dissipative system,Singularity,Convex set,Numerical analysis,Smoothness,Mathematics
Journal
Volume
ISSN
Citations 
385
0021-9991
0
PageRank 
References 
Authors
0.34
0
4
Name
Order
Citations
PageRank
Chenghua Duan100.34
Chun Liu224.84
Cheng Wang35811.05
Xingye Yue451.47