Parallel finite element Laplace transform method for the non-equilibrium groundwater transport equation

Groundwater transport of contaminants undergoing rate-limited or non-equilibrium sorption onto the solid matrix is often described by the dual-porosity or two-domain model, whereby the rate-limited reaction occurs between a mobile and an immobile region. When the sorption reaction is represented by a first-order kinetic relationship, the equation takes the form of a convection–dispersion partial differential equation with an integral term describing the mass transfer between the two regions. An efficient solution algorithm for this type of problems consists in the transformation of the original equation into the Laplace space and subsequent numerical solution of the resulting steady-state equation in the complex space. The exploitation of the Laplace transform to solve the time dependency restricts the application of the technique to linear advection and dispersion terms, while non-linear reactions can be accommodated in particular cases only. This approach has the advantage that it is easily parallelizable, and is therefore proposed in this paper as an efficient algorithm for the parallelization in time of these types of integrodifferential equations. The parallel efficiency of PFELT has been tested on a Cray T3D parallel computer for three sample problems of size N=1071, 3721, and 15 275, respectively, where N is the number of nodal mesh points. The speed-ups obtained vary from 1·98, with two processors, to 39·93 with 64 processors, for the most favorable case. This corresponds to a percentage of parallel work greater than 98 per cent, and a parallel efficiency of more than 60 per cent in the best case, showing the good performance achievable with this algorithm