Dynamical behavior of a nondiffusive scheme for the advection equation

TL;DR

This paper analyzes the long-term behavior of a nondiffusive numerical scheme for the 1D transport equation, showing error bounds, overcompressive behavior at a specific Courant number, and plateau formation.

Contribution

It provides a rigorous analysis of the scheme's overcompressive behavior and plateau stability, extending understanding of its long-term dynamics.

Findings

Error remains uniformly bounded over time.

Scheme becomes overcompressive at Courant number 1/2.

Plateaus form and are stable in certain cases.

Abstract

We study the long time behaviour of a dynamical system strongly linked to the anti-diffusive scheme of Despr\'es and Lagoutiere for the $1$-dimensional transport equation. This scheme is nondiffusive in the sens that discontinuities are not smoothened out through time. Numerical simulations indicates that the scheme error's is uniformly bounded with time. We prove that this scheme is overcompressive when the Courant--Friedrichs--Levy number is 1/2: when the initial data is nondecreasing, the approximate solution becomes a Heaviside function. In a special case, we also understand how plateaus are formed in the solution and their stability, a distinctive feature of the Despr\'es and Lagoutiere scheme.