A second order accurate numerical scheme for the porous medium equation by an energetic variational approach

TL;DR

This paper develops and analyzes a second order accurate numerical scheme for the porous medium equation using an energetic variational approach, ensuring energy stability and convergence.

Contribution

It introduces a novel second order scheme for PME with rigorous stability and convergence analysis based on convexity and asymptotic expansion techniques.

Findings

The scheme is energy stable and uniquely solvable.

Convergence order is optimal based on error estimates.

Numerical examples confirm theoretical results.

Abstract

The porous medium equation (PME) is a typical nonlinear degenerate parabolic equation. An energetic variational approach has been studied in a recent work [6], in which the trajectory equation is obtained, and a few first order accurate numerical schemes have been developed and analyzed. In this paper, we construct and analyze a second order accurate numerical scheme in both time and space. The unique solvability, energy stability are established, based on the convexity analysis. In addition, we provide a detailed convergence analysis for the proposed numerical scheme. A careful higher order asymptotic expansion is performed and two step error estimates are undertaken. In more details, a rough estimate is needed to control the highly nonlinear term in a discrete $W^{1,\infty}$ norm, and a refined estimate is applied to derive the optimal error order. Some numerical examples are presented as well.