An oscillation free local discontinuous Galerkin method for nonlinear degenerate parabolic equations

TL;DR

This paper introduces an oscillation-free local discontinuous Galerkin method tailored for nonlinear degenerate parabolic equations, effectively controlling oscillations while maintaining high accuracy.

Contribution

The paper develops a novel OFLDG method that incorporates damping terms to suppress oscillations in solutions with large gradients, with proven stability and error estimates.

Findings

Maintains high-order accuracy in numerical solutions.

Effectively suppresses spurious oscillations.

Proven $L^2$-stability and optimal error estimates.

Abstract

In this paper, we develop an oscillation free local discontinuous Galerkin (OFLDG) method for solving nonlinear degenerate parabolic equations. Following the idea of our recent work [J. Lu, Y. Liu, and C.-W. Shu, SIAM J. Numer. Anal. 59(2021), pp. 1299-1324.], we add the damping terms to the LDG scheme to control the spurious oscillations when solutions have a large gradient. The $L^2$-stability and optimal priori error estimates for the semi-discrete scheme are established. The numerical experiments demonstrate that the proposed method maintains the high-order accuracy and controls the spurious oscillations well.