The Laplace Transform Galerkin Technique: A time‐continuous finite element theory and application to mass transport in groundwater

Abstract

A time‐continuous numerical technique, referred to as the Laplace Transform Galerkin (LTG) method, is developed and applied to the problem of solute transport in porous media. After application of Galerkin's procedure and subdivision of the domain into finite elements, the method involves a simple application of the Laplace transformation to eliminate the temporal derivatives appearing in the space‐discretized set of ordinary differential equations. Then, by solving the resulting transformed system of algebraic equations in Laplace p space, numerical inversion of the Laplace‐transformed nodal concentration is performed using the robust and accurate Crump (1976) algorithm. The Crump algorithm permits the concentration to be evaluated from a range of time values from a single set of Laplace p space solutions. Because each of the needed p space solutions are independent, the algorithm is well suited for execution on multiprocessor parallel computers. It is demonstrated by means of a series of examples that the LTG scheme is capable of providing highly accurate solutions essentially devoid of numerical dispersion for grid Peclet numbers in excess of 30. Examination of the complex‐valued, Laplace‐domain concentration profiles reveal that they are generally smooth, well‐behaved oscillatory functions compared to the profiles in the time domain, thus permitting the use of a coarse finite element grid. Because of the nature of the Laplace transformation, the LTG method is particularly well suited to the problem of transient groundwater flow and solute transport in fractured porous media or multiple aquifer‐aquitard systems based on the dual‐porosity integrodifferential equation approach.