Iterative schemes for coupled flow and transport in porous media -- Convergence and truncation errors

Nicolae Suciu; Florin A.  Radu; Emil Cătinaş

doi:10.33993/jnaat531-1429

Authors

Nicolae Suciu Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania https://orcid.org/0000-0001-7436-8463
Florin A. Radu Center for Modeling of Coupled Subsurface Dynamics, University of Bergen, Norway https://orcid.org/0000-0002-2577-5684
Emil Cătinaş Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania https://orcid.org/0000-0002-1629-5267

DOI:

https://doi.org/10.33993/jnaat531-1429

Keywords:

Richards' equation, Coupled flow and transport, Finite differences, Global random walk, Iterative schemes, Convergence order

Abstract views: 54

Abstract

Nonlinearities of coupled flow and transport problems for partially saturated porous media are solved with explicit iterative L-schemes. Their behavior is analyzed with the aid of the computational orders of convergence. This approach allows highlighting the influence of the truncation errors in the numerical schemes on the convergence of the iterations. Further, by using manufactured exact solutions, error-based orders of convergence of the iterative schemes are assessed and the convergence of the numerical solutions is demonstrated numerically through grid-convergence tests.

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References

C.D. Alecsa, I. Boros, F. Frank, P. Knabner, M. Nechita, A. Prechtel, A. Rupp and N. Suciu, Numerical benchmark study for fow in heterogeneous aquifers, Adv. Water Resour., 138 (2020), 103558. https://doi.org/10.1016/j.advwatres.2020.103558 DOI: https://doi.org/10.1016/j.advwatres.2020.103558

E. Cătinaș, A survey on the high convergence orders and computational convergence orders of sequences, Appl. Math. Comput. 343 (2019), pp. 1-20. https://doi.org/10.1016/j.amc.2018.08.006 DOI: https://doi.org/10.1016/j.amc.2018.08.006

E. Cătinaș, How many steps still left to x*?, SIAM Rev. 63 (2021) no. 3, pp. 585-624. https://doi.org/10.1137/19M1244858 DOI: https://doi.org/10.1137/19M1244858

D. Illiano, I.S. Pop and F.A. Radu, Iterative schemes for surfactant transport in porous media, Comput. Geosci. 25 (2021), 805–822. https://doi.org/10.1007/s10596-020-09949-2 DOI: https://doi.org/10.1007/s10596-020-09949-2

P. Knabner, S. Bitterlich, R. Iza Teran, R., A. Prechtel and E. Schneid, Influence of surfactants on spreading of contaminants and soil remediation, pp.152-161 In: J ̈ager, W., Krebs, H.J. (Eds.), Mathematics-Key Technology for the Future, Springer, Berlin, Heidelberg, 2003. https://doi.org/10.1007/978-3-642-55753-8_12 DOI: https://doi.org/10.1007/978-3-642-55753-8_12

P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, Nwe York, 2003. https://doi.org/10.1007/b97419 DOI: https://doi.org/10.1007/b97419

F. List and F.A. Radu, A study on iterative methods for solving Richards’ equation, Comput. Geosci. 20 (2016) no. (2), 341-353. https://doi.org/10.1007/s10596-016-9566-3 DOI: https://doi.org/10.1007/s10596-016-9566-3

I.S. Pop, F.A. Radu and P. Knabner, Mixed finite elements for the Richards’ equation: linearization procedure, J. Comput. Appl. Math. 168 (2004) no. 1, pp. 365-373 https://doi.org/10.1016/j.cam.2003.04.008 DOI: https://doi.org/10.1016/j.cam.2003.04.008

F.A. Radu, K. Kumar, J.M. Nordbotten and I.S. Pop, A robust, mass conservative scheme for two-phase flow in porous media including H ̈older continuous nonlinearities, IMA Journal of Numerical Analysis, 38 (2018) no. 2, pp.884-920. https://doi.org/10.1093/imanum/drx032 DOI: https://doi.org/10.1093/imanum/drx032

P.J. Roache, Code verification by the method of manufactured solutions, J. Fluids Eng. 124 (2002) no. 1, pp. 4-10. https://dx.doi.org/10.1115/1.1436090 DOI: https://doi.org/10.1115/1.1436090

C.J. Roy, Review of code and solution verification procedures for compu- tational simulation, J. Comput. Phys. 205 (2005), 131-156. https://doi.org/10.1016/j.jcp.2004.10.036 DOI: https://doi.org/10.1016/j.jcp.2004.10.036

J.S. Stokke, K. Mitra, E. Storvik, J.W. Both and F.A. Radu, An adaptive solution strategy for Richards’ equation, Comput. Math. Appl. 152 (2023), 155-167. https://doi.org/10.1016/j.camwa.2023.10.020 DOI: https://doi.org/10.1016/j.camwa.2023.10.020

J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, 2004. https://doi.org/10.1137/1.9780898717938 DOI: https://doi.org/10.1137/1.9780898717938

N. Suciu, Diffusion in Random Fields. Applications to Transport in Groundwater, Birkhauser, Cham, 2019. https://doi.org/10.1007/978-3-030-15081-5 DOI: https://doi.org/10.1007/978-3-030-15081-5

N. Suciu, Global Random Walk Solutions for Flow and Transport in Porous Media, in F.J. Vermolen and C. Vuik (eds.), Numerical Mathematics and Advanced Applications ENUMATH 2019, Lecture Notes in Computational Science and Engineering 139, Springer Nature, Switzerland, 2020. https://doi.org/10.1007/978-3-030-55874-1_93 DOI: https://doi.org/10.1007/978-3-030-55874-1_93

N. Suciu, D. Illiano, A. Prechtel and F.A. Radu, Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media, Adv. Water Resour. 152 (2021), 103935. https://doi.org/10.1016/j.advwatres.2021.103935 DOI: https://doi.org/10.1016/j.advwatres.2021.103935

N. Suciu and F.A. Radu, Global random walk solvers for reactive transport and biodegradation processes in heterogeneous porous media, Adv. Water Res., 166 (2022), 104268. https://doi.org/10.1016/j.advwatres.2022.104268 DOI: https://doi.org/10.1016/j.advwatres.2022.104268

N. Suciu, F.A. Radu and I.S. Pop, Space-time upscaling of reactive transport in porous media, Adv. Water Resour. 176 (2023), 104443. https://dx.doi.org/10.1016/j.advwatres.2023.104443 DOI: https://doi.org/10.1016/j.advwatres.2023.104443

N. Suciu, F.A. Radu, J.S. Stokke, E. Catinas ̧ and A. Malina, Computational orders of convergence of iterative methods for Richards’ equation, arXiv preprint arXiv:2402.00194 (2004), https://doi.org/10.48550/arXiv.2402.00194

C. Vamoș, N. Suciu and M Peculea, Numerical modelling of the one-dimensional diffusion by random walkers, Rev. Anal. Numer. Theor. Approx., 26 (1997) no. 1-2, pp. 237–247. https://ictp.acad.ro/jnaat/journal/article/view/1997-vol26-nos1-2-art32/

C. Vamoș, N. Suciu and H. Vereecken, Generalized random walk algorithm for the numerical modeling of complex diffusion processes, J. Comput. Phys., 186 (2003), pp. 527–544. https://doi.org/10.1016/S0021-9991(03)00073-1 DOI: https://doi.org/10.1016/S0021-9991(03)00073-1