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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