TY - JOUR

T1 - Numerical methods for porous medium equation by an energetic variational approach

AU - Duan, Chenghua

AU - Liu, Chun

AU - Wang, Cheng

AU - Yue, Xingye

N1 - Funding Information:
This work is supported in part by NSF of China under the grants 11271281 . Chun Liu and Cheng Wang are partially supported by NSF grants DMS-1216938 , DMS-1418689 , respectively.
Funding Information:
This work is supported in part by NSF of China under the grants 11271281. Chun Liu and Cheng Wang are partially supported by NSF grants DMS-1216938, DMS-1418689, respectively.
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2019/5/15

Y1 - 2019/5/15

N2 - 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 flogf 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.

AB - 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 flogf 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.

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U2 - 10.1016/j.jcp.2019.01.055

DO - 10.1016/j.jcp.2019.01.055

M3 - Article

AN - SCOPUS:85062152756

VL - 385

SP - 13

EP - 32

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -