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High Order Whitney Forms on Simplices and the Question of Potentials Read chapter Read first chapter

2021 | OriginalPaper | Chapter

A New Algebraically Stabilized Method for Convection–Diffusion–Reaction Equations

Author : Petr Knobloch

Published in: Numerical Mathematics and Advanced Applications ENUMATH 2019

Publisher: Springer International Publishing

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Abstract

This paper is devoted to algebraically stabilized finite element methods for the numerical solution of convection–diffusion–reaction equations. First, the algebraic flux correction scheme with the popular Kuzmin limiter is presented. This limiter has several favourable properties but does not guarantee the validity of the discrete maximum principle for non-Delaunay meshes. Therefore, a generalization of the algebraic flux correction scheme and a modification of the limiter are proposed which lead to the discrete maximum principle for arbitrary meshes. Numerical results demonstrate the advantages of the new method.

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Literature
1.
Gabriel R. Barrenechea, Volker John, and Petr Knobloch. Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension. IMA J. Numer. Anal., 35(4):1729–1756, 2015.MathSciNetCrossRef
2.
Gabriel R. Barrenechea, Volker John, and Petr Knobloch. Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal., 54(4):2427–2451, 2016.MathSciNetCrossRef
3.
Gabriel R. Barrenechea, Volker John, Petr Knobloch, and Richard Rankin. A unified analysis of algebraic flux correction schemes for convection-diffusion equations. SeMA J., 75(4):655–685, 2018.MathSciNetCrossRef
4.
Petr Knobloch. On the discrete maximum principle for algebraic flux correction schemes with limiters of upwind type. In Z. Huang, M. Stynes, and Z. Zhang, editors, Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016, volume 120 of Lecture Notes in Computational Science and Engineering, pages 129–139. Springer, 2017.
5.
Dmitri Kuzmin. Algebraic flux correction for finite element discretizations of coupled systems. In M. Papadrakakis, E. Oñate, and B. Schrefler, editors, Proceedings of the Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering, pages 1–5. CIMNE, Barcelona, 2007.
6.
Dmitri Kuzmin. Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. J. Comput. Appl. Math., 236:2317–2337, 2012.MathSciNetCrossRef
7.
Dmitri Kuzmin and Matthias Möller. Algebraic flux correction I. Scalar conservation laws. In Dmitri Kuzmin, Rainald Löhner, and Stefan Turek, editors, Flux-Corrected Transport. Principles, Algorithms, and Applications, pages 155–206. Springer-Verlag, Berlin, 2005.
Metadata
Title
A New Algebraically Stabilized Method for Convection–Diffusion–Reaction Equations
Author
Petr Knobloch
Copyright Year
2021
Book
Numerical Mathematics and Advanced Applications ENUMATH 2019

Print ISBN: 978-3-030-55873-4

Electronic ISBN: 978-3-030-55874-1

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-55874-1_59

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