Numerical exploration of a forward-backward diffusion equation

TL;DR

This paper numerically investigates a forward-backward diffusion equation with a cubic-like diffusion function, focusing on its entropy formulation and comparing second and third order schemes to analyze interface propagation.

Contribution

It introduces numerical schemes for both second and third order formulations of the equation and compares their effectiveness in modeling phase transition interfaces.

Findings

Third order scheme captures interface propagation more accurately.

Second order scheme is computationally simpler but less precise.

Comparison reveals strengths and limitations of each approach.

Abstract

We analyze numerically a forward-backward diffusion equation with a cubic-like diffusion function, -emerging in the framework of phase transitions modeling- and its "entropy" formulation determined by considering it as the singular limit of a third-order pseudo-parabolic equation. Precisely, we propose schemes for both the second and the third order equations, we discuss the analytical properties of their semi-discrete counter-parts and we compare the numerical results in the case of initial data of Riemann type, showing strengths and flaws of the two approaches, the main emphasis being given to the propagation of transition interfaces.