Large-time Behavior of the Solutions to Rosenau Type Approximations to the Heat Equation

TL;DR

This paper investigates how solutions to Rosenau type approximations to the heat equation behave over long times, revealing convergence to the fundamental solution at a sub-optimal rate, including for central differences schemes.

Contribution

It demonstrates for the first time that Rosenau type approximations, including central differences schemes, approach the heat equation's fundamental solution over time.

Findings

Solutions approach the fundamental heat solution at a sub-optimal rate.

The convergence property applies to central differences schemes.

First observation of this behavior in such schemes.

Abstract

In this paper we study the large-time behavior of the solution to a general Rosenau type approximation to the heat equation, by showing that the solution to this approximation approaches the fundamental solution of the heat equation at a sub-optimal rate. The result is valid in particular for the central differences scheme approximation of the heat equation, a property which to our knowledge has never been observed before.