Title :

Uniform estimates for a fully discrete scheme integrating the linear heat equation on a bounded interval with pure Neumann boundary conditions

Author(s) :

Dujardin, Guillaume [Auteur]
Systèmes de particules et systèmes dynamiques [Paradyse]
Lafitte, Pauline [Auteur]
Mathématiques et Informatique pour la Complexité et les Systèmes [MICS]
Fédération de Mathématiques de l'Ecole Centrale Paris [FR3487]

Publication date :

2023-11-24

HAL domain(s) :

Mathématiques [math]/Mathématiques générales [math.GM]

English abstract : [en]

This manuscript deals with the analysis of numerical methods for the full discretization (in time and space) of the linear heat equation with Neumann boundary conditions, and it provides the reader with error estimates ...
Show more >This manuscript deals with the analysis of numerical methods for the full discretization (in time and space) of the linear heat equation with Neumann boundary conditions, and it provides the reader with error estimates that are uniform in time. First, we consider the homogeneous equation with homogeneous Neumann boundary conditions over a finite interval. Using finite differences in space and the Euler method in time, we prove that our method is of order 1 in space, uniformly in time, under a classical CFL condition, and despite its lack of consistency at the boundaries. Second, we consider the nonhomogeneous equation with nonhomogeneous Neumann boundary conditions over a finite interval. Using a tailored similar scheme, we prove that our method is also of order 1 in space, uniformly in time, under a classical CFL condition. We indicate how this numerical method allows for a new way to compute steady states of such equations when they exist. We conclude by several numerical experiments to illustrate the sharpness and relevance of our theoretical results, as well as to examine situations that do not meet the hypotheses of our theoretical results, and to illustrate how our results extend to higher dimensions.Show less >

Language :

Anglais

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL