Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension

Barrenechea, Gabriel and Volker, John and Knobloch, Petr (2014) Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension. IMA Journal of Numerical Analysis. ISSN 0272-4979 (https://doi.org/10.1093/imanum/dru041)

[thumbnail of Barrenechea-etal-IMAJNA-2014-Some-analytical-results-for-an-algebraic-flux-correction-scheme]
Preview
PDF. Filename: Barrenechea_etal_IMAJNA_2014_Some_analytical_results_for_an_algebraic_flux_correction_scheme.pdf
Accepted Author Manuscript

Download (2MB)| Preview

Abstract

Algebraic flux correction schemes are nonlinear discretizations of convection dominated problems. In this work, a scheme from this class is studied for a steady-state convection–diffusion equation in one dimension. It is proved that this scheme satisfies the discrete maximum principle. Also, as it is a nonlinear scheme, the solvability of the linear subproblems arising in a Picard iteration is studied, where positive and negative results are proved. Furthermore, the non-existence of solutions for the nonlinear scheme is proved by means of counterexamples. Therefore, a modification of the method, which ensures the existence of a solution, is proposed. A weak version of the discrete maximum principle is proved for this modified method.