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

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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)

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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.

ORCID iDs

Barrenechea, Gabriel ORCID logoORCID: https://orcid.org/0000-0003-4490-678X, Volker, John and Knobloch, Petr;
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