Communications in Mathematical Sciences

Volume 20 (2022)

Number 4

A second-order accurate, energy stable numerical scheme for the one-dimensional porous medium equation by an energetic variational approach

Pages: 978 – 1024



Chenghua Duan (Department of Mathematics, Shanghai University, Shanghai, China; and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China)

Wenbin Chen (School of Mathematical Sciences, Fudan University, Shanghai, China)

Chun Liu (Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Il., U.S.A.)

Cheng Wang (Department of Mathematics, University of Massachusetts, North Dartmouth, Mass., U.S.A.)

Xingye Yue (Department of Mathematics, Soochow University, Suzhou, China)


The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation. An energetic variational approach (EVA) provides many insights to such a physical model, in which the trajectory equation can be obtained, based on different dissipative energy laws. In this article, we propose and analyze a second-order accurate-in-time numerical scheme for the PME in the EVA approach. A modified Crank–Nicolson temporal discretization is applied, combined with the finite difference over a uniform spatial mesh. Such a numerical scheme is highly nonlinear, and it is proved to be uniquely solvable on an admissible convex set, in which the convexity of the nonlinear implicit terms will play an important role. Subsequently, the energy dissipation property is established, with careful summation by parts formulas applied in the spatial discretization. More importantly, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to fourth-order temporal and spatial accuracy), the rough error estimate (to establish the $W^{1,\infty}_h$ bound for the numerical variable), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, it will be the first work to combine three theoretical properties for a second order accurate numerical scheme to the PME in the EVA approach: unique solvability, energy stability and optimal rate convergence analysis. A few numerical results are also presented in this article, which demonstrate the robustness of the proposed numerical scheme.


energetic variational approach, porous medium equation, trajectory equation, optimal rate convergence analysis

2010 Mathematics Subject Classification

35K35, 35K55, 49J40, 65M06, 65M12

Received 13 August 2020

Received revised 15 October 2021

Accepted 15 October 2021

Published 11 April 2022