Laplace transform analytic element method for transient flow problems

Recently we [12] proposed a new Laplace transform analytic element method (LT-AEM) for the solution of transient groundwater flow problems. Laplace transformation of the transient flow equations leads to a time-independent modified Helmholtz equation and associated conditions at discontinuities. The latter are solved by AEM and the results transformed numerically back into the time domain. Though neither continuity of potential nor continuity of flux are satisfied automatically at internal discontinuities, both are satisfied approximately via least squares at an overdetermined system of control points in a manner similar to that of Janković [17] and Barnes and Janković [5]. LT-AEM preserves all advantages of the AEM in Laplace space, most importantly its mathematical elegance and grid-free nature. Solution in Laplace space and numerical back transformation into the time domain are done independently for any given time and are thus amenable to parallel computation on multiple processors. This renders the method particularly well suited for cases where a high-accuracy solution is required at a relatively small number of discrete space-time locations. LT-AEM requires a new family of analytic elements associated transformation of known analytic solutions in the time domain, or developed directly in the Laplace domain. We use both methods to develop a number of analytic elements for LT-AEM. We then illustrate the method on transient flow in a two-dimensional confined aquifer containing various inhomogeneities and time-dependent sources.