Review of Inverse Laplace Transform Algorithms for Laplace-Space Numerical Approaches

TL;DR

This paper compares five numerical inverse Laplace transform algorithms using a boundary element method simulation to evaluate their efficiency and robustness in solving Laplace-transformed diffusion equations.

Contribution

It provides a comprehensive comparison of inverse Laplace transform algorithms and discusses implementation techniques to reduce the number of Laplace-space evaluations needed.

Findings

Fourier-series based algorithms are robust and effective.

These algorithms allow re-use of image functions across multiple time points.

They are suitable for common time behaviors in Laplace-space numerical methods.

Abstract

A boundary element method (BEM) simulation is used to compare the efficiency of numerical inverse Laplace transform strategies, considering general requirements of Laplace-space numerical approaches. The two-dimensional BEM solution is used to solve the Laplace-transformed diffusion equation, producing a time-domain solution after a numerical Laplace transform inversion. Motivated by the needs of numerical methods posed in Laplace-transformed space, we compare five inverse Laplace transform algorithms and discuss implementation techniques to minimize the number of Laplace-space function evaluations. We investigate the ability to calculate a sequence of time domain values using the fewest Laplace-space model evaluations. We find Fourier-series based inversion algorithms work for common time behaviors, are the most robust with respect to free parameters, and allow for straightforward image function evaluation re-use across at least a log cycle of time.