Performance Analysis of Stehfest and Power Series Expansion Methods for Solution to Diffusive and Advective Transport Problems

Abstract:

This paper presents results of the test of methods for numerical inversion of the Laplace Transform for solving the one-dimensional advection-diffusion equation, which describes solute transport processes, focusing on the contaminant transport in a porous medium. The performance of Stehfest and Power Series Expansion methods is analyzed, for diffusion-dominated and advection-dominated transport problems under linear flow condition. Numerical results are compared to the analytical solution by means of the absolute error. Based on these results, we concluded that both methods, Stehfest and Power Series Expansion, are recommended only for diffusion-dominated cases.

• Periodical: Defect and Diffusion Forum (Volume 396)
• Pages: 99-108
• DOI: https://doi.org/10.4028/www.scientific.net/DDF.396.99
• Online since: August 2019
• Authors: Cristiana Pilatti*, Bárbara Denicol do Amaral Rodriguez, João Francisco Prolo Filho
• Keywords: Advection-Diffusion Equation, Laplace Transform, Numerical Inversion

References

[1] J. Bear, Dynamics of fluids in porous media, Dover Publications, New York, 1972. Originally published: American Elsevier Publishing Company, New York.

DOI: 10.1016/0160-9327(73)90013-6

[2] J.Abate, P. P. Valkó, Multi-precision Laplace transform inversion, International Journal for Numerical Methods in Engineering. 60(5) (2004) 979-993.

DOI: 10.1002/nme.995

[3] A. M. Cohen, Numerical Methods for Laplace Transform Inversion, Springer Science, New York, (2007).

[4] B. Davies, B. Martin, Numerical inversion of the Laplace transform: a survey and comparison of methods, Journal of Computational Physics. 33(1) (1979) 1-32.

DOI: 10.1016/0021-9991(79)90025-1

[5] D. G. Duffy, On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications, ACM Transactions on Mathematical Software. 19(3) (1993) 333-359.

DOI: 10.1145/155743.155788

[6] H. Stehfest, Algorithm 368: Numerical inversion of Laplace transforms [D5], Communications of the ACM, 13(1) (1970) 47-49.

DOI: 10.1145/361953.361969

[7] H. Stehfest, Remark on algorithm 368: Numerical inversion of Laplace transforms, Communications of the ACM. 13(10) (1970) 624.

DOI: 10.1145/355598.362787

[8] H.Y. Chung, Y.Y. Sun, Taylor series approach to functional approximation for inversion of Laplace transforms, Electronics Letters. 22(23) (1986) 1219-1221.

DOI: 10.1049/el:19860836

[9] A. F. Moench, A. Ogata, A Numerical Inversion of the Laplace TransformSolution to Radial Dispersion in aPorous Medium, Water Resources Research. 17(1) (1981) 250-252.

DOI: 10.1029/wr017i001p00250

[10] C.-S. Chen, Analytical and Approximate Solutions to Radial Dispersion From an Injection Well to a Geological Unit With Simultaneous Diffusion Into Adjacent Strata, Water Resources Research. 21(8) (1985) 1069-1076.

DOI: 10.1029/wr021i008p01069

[11] Q. Wanga, H. Zhan, On different numerical inverse Laplace methods for solute transport problems, Advances in Water Resources. 75 (2015) 80-92.

DOI: 10.1016/j.advwatres.2014.11.001

[12] F. R. De Hoog, J. H. Knight, A. N. Stokes, An improved method for numerical inversion of Laplace transforms, SIAM Journal on Scientific and Statistical Computing. 3(3) (1982) 357-366.

DOI: 10.1137/0903022

[13] G. Honig, U. Hirdes, A method for the numerical inversion of Laplace transforms, Journal of Computational and Applied Mathematics. 10(1) (1984) 113-132.

DOI: 10.1016/0377-0427(84)90075-x

[14] A. Talbot, The accurate numerical inversion of Laplace transforms, IMA Journal of Applied Mathematics. 23(1) (1979) 97-120.

DOI: 10.1093/imamat/23.1.97

[15] W. T. Weeks, Numerical inversion of Laplace transforms using Laguerre functions, Journal of the Association for Computing Machinery. 13(3) (1966) 419-429.

DOI: 10.1145/321341.321351

[16] R. M. Simon, M. T. Stroot, G. H. Weiss, Numerical inversion of Laplace transforms with application to percentage labeled mitoses experiments, Computers and Biomedical Research. 5(6) (1972) 596-607.

DOI: 10.1016/0010-4809(72)90039-0

[17] V. Zakian, Numerical inversion of Laplace transform, Electronics Letters. 5(6) (1969) 120-121.

[18] A. Ogata, R. B. Banks, A Solution of the Differential Equation of Longitudinal Dispersion in Porous Media, Geological survey professional paper 411-A. United States Government Printing Office, Washington, DC, (1961).

DOI: 10.3133/pp411a

[19] E. Kreyszig, Advanced Engineering Mathematics, tenth ed., John Wiley & Sons, Inc., (2011).

[20] D. P. Gaver Jr., Observing Stochastic Processes, and Approximate Transform Inversion, Operations Research. 14(3) (1966) 444-459.

DOI: 10.1287/opre.14.3.444