An Efficient Implementation of the Method of Lines for Multicomponent Reactive Transport Equations Marwan Fahs & Anis Younes & Philippe Ackerer # Springer Science+Business Media B.V. 2010 Abstract Modeling reactive transport with chemical equilibrium reactions requires solution of coupled par- tial differential and algebraic equations. In this work, two formulations are developed to combine the method of lines (MOL) with the global implicit approach. The first formulation has a non-conservative form and leads to a nonlinear system of ordinary differential equations with a reduced number of unknowns. The second formulation presents better conservation properties but leads to a nonlinear system of differential algebraic equations with a large number of unknowns. In both formulations, the resulting systems are integrated in time using the DLSODIS time solver which adapts both the order of the time integration and the time step size to provide the necessary accuracy. Numerical experiments show that higher-order time integration is effective for solving the non-conservative formulation and point out the high benefit of the MOL for solving reactive transport problems. Keywords Reactive transport . Equilibrium reactions . Global implicit approach . Method of lines . Higher-order time integration 1 Introduction In this work, multicomponent reactive transport (MRT) problems are considered with chemical equilibrium reactions. These problems are common in water resour- ces engineering, soil sciences, chemical engineering, and many other fields. Under equilibrium assumption, MRT problems are often described by two sets of equations. The first set of partial differential equations (PDEs) describes the transport of the chemical components. The second is a set of nonlinear algebraic equations (AEs) describing the chemical reactions among the components them- selves. Thus, the simulation of MRT problems re- quires the solution of a coupled system of PDEs and AEs. This system is strongly nonlinear, and the total number of equations can be quite large. Therefore, its numerical solution requires a large amount of com- putational time even on fast computers. Several approaches have been developed for the simulation of MRT problems over the last decades. These approaches are based upon the operator splitting (OS) approach or the global implicit approach (GIA). In the OS approach, chemical and transport equations are solved separately (Yeh and Tripathi 1989; Engesgaard and Kipp 1992; Miller and Rabideau 1993; Steefel and DOI 10.1007/s11270-010-0477-y M. Fahs School of Engineering, Lebanese International University, Beirut, Lebanon A. Younes (*) : P. Ackerer Laboratoire d’Hydrologie et de Géochimie de Strasbourg, Université de Strasbourg/EOST, CNRS, 1 rue de Blessig, 67000 Strasbourg, France e-mail: younes@unistra.fr Water Air Soil Pollut (2011) 215:273–283 Received: 6 January 2010 / Accepted: 12 May 2010 / Published online: 29 May 2010