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

Abstract

The Laplace Transform Boundary Element (LTBE) method is a recently introduced numerical method, and has been used for the solution of diffusion-type PDEs. It completely eliminates the time dependency of the problem and the need for time discretization, yielding solutions numerical in space and semi-analytical in time. In LTBE solutions are obtained in the Laplace space, and are then inverted numerically to yield the solution in time. The Stehfest and the DeHoog formulations of LTBE, based on two different inversion algorithms, are investigated. Both formulations produce comparable, extremely accurate solutions. The Stehfest formulation uses real values for the Laplace space parameter λ, combines linearly the results of a limited number of matrix solutions (6 to 8), does not increase computer storage, is simple to code, and requires significantly less execution time, but yields a solution at a single observation time t for each set of λ’s. The DeHoog formulation uses complex values for the λ’s, needs more matrix inversions, and uses non-linear combinations of the solutions, but allows solutions at a range of times t from a single set of λ’s. Compared to the Stehfest LTBE, the DeHoog LTBE produces matrices 4 times as large, increases execution times per matrix inversion by at least a factor of 12 and the memory requirements by a minimum factor of 4. The Stehfest LTBE seems to have a clear advantage, except in cases involving very steep functions of time.KeywordsBoundary ElementBoundary Element MethodNumerical InversionRoundoff ErrorLaplace SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.