A numerical study of an Heaviside function driven degenerate diffusion equation

TL;DR

This paper investigates a nonlinear degenerate parabolic equation driven by a Heaviside function, analyzing its behavior as an evolutive variational inequality and exploring a finite difference numerical method with results.

Contribution

It introduces a novel model where the diffusion depends on the solution's distance from a target, and studies its theoretical properties and numerical solutions.

Findings

Solution evolves towards the target over time

Model behaves as an evolutive variational inequality

Finite difference method effectively approximates the solution

Abstract

We analyze a nonlinear degenerate parabolic problem whose diffusion coefficient is the Heaviside function of the distance of the solution itself from a given target function. We show that this model behaves as an evolutive variational inequality having the target as an obstacle: under suitable hypotheses, starting from an initial state above the target the solution evolves in time towards an asymptotic solution, eventually getting in contact with part of the target itself. We also study a finite difference approach to the solution of this problem, using the exact Heaviside function or a regular approximation of it, showing the results of some numerical tests.