### Abstract

Conservation laws of advection-diffusion-reaction (ADR) type are ubiquitous in continuum physics. In this paper we outline discretization of these problems and iterative schemes for the resulting linear system. For discretization we use the finite volume method in combination with the complete flux scheme. The numerical flux is the superposition of a homogeneous flux, corresponding to the advection-diffusion operator, and the inhomogeneous flux, taking into account the effect of the source term (ten Thije Boonkkamp and Anthonissen, J Sci Comput 46(1):47–70, 2011). For a three-dimensional conservation law this results in a 27-point coupling for the unknown as well as the source term. Direct solution of the sparse linear systems that arise in 3D ADR problems is not feasible due to fill-in. Iterative solution of such linear systems remains to be the only efficient alternative which requires less memory and shorter time to solution compared to direct solvers. Iterative solvers require a preconditioner to reduce the number of iterations. We present results using several different preconditioning techniques and study their effectiveness.

Original language | English |
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Title of host publication | Numerical Mathematics and Advanced Applications ENUMATH 2015 |

Editors | B. Karasözen, M. Manguoğlu, M. Tezer-Sezgin, S. Göktepe, Ö. Uğur |

Place of Publication | Dordrecht |

Publisher | Springer |

Pages | 275-283 |

Number of pages | 9 |

ISBN (Electronic) | 978-3-319-39929-4 |

ISBN (Print) | 978-3-319-39927-0 |

DOIs | |

Publication status | Published - 2016 |

Event | 2015 European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2015) - Middle East Technical University, Ankara, Turkey Duration: 14 Sep 2015 → 18 Sep 2015 |

### Publication series

Name | Lecture Notes in Computational Science and Engineering |
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Volume | 112 |

ISSN (Print) | 14397358 |

### Conference

Conference | 2015 European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2015) |
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Abbreviated title | ENUMATH 2015 |

Country | Turkey |

City | Ankara |

Period | 14/09/15 → 18/09/15 |

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## Cite this

*Numerical Mathematics and Advanced Applications ENUMATH 2015*(pp. 275-283). (Lecture Notes in Computational Science and Engineering; Vol. 112). Dordrecht: Springer. https://doi.org/10.1007/978-3-319-39929-4_27