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Computational Science, Engineering & Technology Series
ISSN 1759-3158
Edited by: B.H.V. Topping and P. Iványi
Chapter 7

Solving Partial Differential Algebraic Equations and Reactive Transport Models

J. Erhel1, S. Sabit1 and C. de Dieuleveult2

1Inria, Rennes, France
2Mines ParisTech, Fontainebleau, France

Full Bibliographic Reference for this chapter
J. Erhel, S. Sabit, C. de Dieuleveult, "Solving Partial Differential Algebraic Equations and Reactive Transport Models", in B.H.V. Topping and P. Iványi, (Editor), "Developments in Parallel, Distributed, Grid and Cloud Computing for Engineering", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 7, pp 151-169, 2013. doi:10.4203/csets.31.7
Keywords: partial differential algebraic equations, Newton, fixed-point, reactive transport.

In some scientific applications, such as groundwater studies, several processes are represented by coupled models. For example, a density-driven flow model couples the flow equations with the transport of salt. A reactive transport model couples transport equations of pollutants with chemical equations. The coupled model can combine partial differential equations with algebraic equations, in a so-called PDAE system, which is in general nonlinear. A classical approach is to follow a method of lines, where space is first discretized, leading to a semi-discrete differential algebraic system (DAE). Then time is discretized by a scheme tuned for DAE, such that each time step requires the solving of a nonlinear system of equations. In some decoupled approaches, a fixed-point technique is used. However, a Newton method converges faster in general and is more efficient, even though each iteration is more CPU-intensive. In this chapter, we deal with reactive transport models and show how a Newton method can be used efficiently. Numerical experiments illustrate the efficiency of a substitution technique. Moreover, it appears that using logarithms in the chemistry equations lead to ill conditioned matrices and increase the computational cost.

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