The Laplace Transform Boundary Element (LTBE) Method for the Solution of Diffusion-Type Equations

Abstract

A new numerical method, the Laplace Transform Boundary Element (LTBE) method, was developed for the solution of diffusion-type PDE’s by eliminating the time dependency of the problem using a Laplace transform formulation. In essence, LTBE yields a solution numerical in space and semi-analytical in time. It consists of 4 steps: 1) a Laplace transform is performed on the PDE, 2) the transformed PDE is approximated using the Boundary Element Method (BEM), 3) the resulting system of simultaneous equations is solved and the transformed vector of unknowns is determined in the Laplace space, and 4) the solution vector obtained in step 3 is inverted numerically to yield the solution in time. The solution in the Laplace space renders the time interpolation and time marching schemes employed in the standard BEM irrelevant because time is no longer considered. The method was tested against results obtained from two- and three-dimensional test cases of heat conduction and groundwater flow using a standard BEM simulator and analytical models. For a single time-step, LTBE requires an execution time 6 times longer than the analogous BEM requirement without an increase in storage. This disadvantage is outweighed by the fact that LTBE allows an unlimited time-step size with a more accurate solution than the conventional BEM treatment. Execution times are reduced by orders of magnitude because calculations in the LTBE scheme are necessary only at the desired observation times, while the standard BEM requires calculations at all the intermediate times of the discretized time domain. Roundoff and material balance errors are stable, non-increasing, and much lower than in the standard BEM simulations.