Two stable methods with numerical experiments for solving the backward heat equation

Authors:
Fabien Ternat;Oscar Orellana;Prabir Daripa
Affiliations:
Institut de Recherche sur les Phénomènes Hors-íquilibre, IRPHE, Marseille, France;Departamento de matemáticas, Universidad Técnica Santa María de Valparaíso, UTFSM, Chile;Department of Mathematics, Texas A&M University, College Station, TX, United States
Venue:
Applied Numerical Mathematics
Year:
2011

Citing 5
Cited 1

Optimal filtering for the backward heat equation

SIAM Journal on Numerical Analysis
Some useful filtering techniques for illposed problems7

Journal of Computational and Applied Mathematics
A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: filtering and regularization techniques

Applied Mathematics and Computation
A non-overlap domain decomposition method for the forward-backward heat equation

Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 6th Japan--China joint seminar on numerical mathematics, university of Tsukuba, Japan, 5-9 August 2002
Regularization Parameter Selection in Discrete Ill-Posed Problems-The Use of the U-Curve

International Journal of Applied Mathematics and Computer Science

Total variation regularization for the reconstruction of a mountain topography

Applied Numerical Mathematics

Quantified Score

Hi-index	0.00

Visualization

Abstract

This paper presents results of some numerical experiments on the backward heat equation. Two quasi-reversibility techniques, explicit filtering and structural perturbation, to regularize the ill-posed backward heat equation have been used. In each of these techniques, two numerical methods, namely Euler and Crank-Nicolson (CN), have been used to advance the solution in time. Crank-Nicolson method is very counter-intuitive for solving the backward heat equation because the dispersion relation of the scheme for the backward heat equation has a singularity (unbounded growth) for a particular wave whose finite wave number depends on the numerical parameters. In comparison, the Euler method shows only catastrophic growth of relatively much shorter waves. Strikingly we find that use of smart filtering techniques with the CN method can give as good a result, if not better, as with the Euler method which is discussed in the main text. Performance of these regularization methods using these numerical schemes have been exemplified.