Semi-explicit integration of second order for weakly coupled poroelasticity. (English) Zbl 07837843
Summary: The paper investigates a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition.
Overall, the paper is good and I recommand for publication.
Overall, the paper is good and I recommand for publication.
Reviewer: R. Ishwariya (Chennai)
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65L80 | Numerical methods for differential-algebraic equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74L05 | Geophysical solid mechanics |
| 35A24 | Methods of ordinary differential equations applied to PDEs |
Keywords:
poroelasticity; elliptic-parabolic problem; semi-explicit time discretization; delay; backward differentiation formulaSoftware:
RODASReferences:
| [1] | Akrivis, G.; Crouzeix, M.; Makridakis, C., Implicit-explicit multistep finite element methods for nonlinear parabolic problems, Math. Comp., 67, 222, 457-477, 1998 · Zbl 0896.65066 · doi:10.1090/S0025-5718-98-00930-2 |
| [2] | Akrivis, G.; Lubich, C., Fully implicit, linearly implicit and implicit-explicit backward difference formulae for quasi-linear parabolic equations, Numer. Math., 131, 713-735, 2015 · Zbl 1334.65124 · doi:10.1007/s00211-015-0702-0 |
| [3] | Altmann, R.; Chung, E.; Maier, R.; Peterseim, D.; Pun, SM, Computational multiscale methods for linear heterogeneous poroelasticity, J. Comput. Math., 38, 1, 41-57, 2020 · Zbl 1463.65284 · doi:10.4208/jcm.1902-m2018-0186 |
| [4] | Altmann, R.; Maier, R., A decoupling and linearizing discretization for poroelasticity with nonlinear permeability, SIAM J. Sci. Comput., 44, 3, B457-B478, 2022 · Zbl 1496.65153 · doi:10.1137/21M1413985 |
| [5] | Altmann, R.; Maier, R.; Unger, B., Semi-explicit discretization schemes for weakly-coupled elliptic-parabolic problems, Math. Comp., 90, 329, 1089-1118, 2021 · Zbl 1466.65115 · doi:10.1090/mcom/3608 |
| [6] | Altmann, R., Zimmer, C.: On the smoothing property of linear delay partial differential equations. J. Math. Anal. Appl. 467(2), 916-934 (2018). doi:10.1016/j.jmaa.2018.07.049 · Zbl 1395.35058 |
| [7] | Armero, F.; Simo, JC, A new unconditionally stable fractional step method for nonlinear coupled thermomechanical problems, Internat. J. Numer. Methods Engrg., 35, 4, 737-766, 1992 · Zbl 0784.73085 · doi:10.1002/nme.1620350408 |
| [8] | Bellen, A., Zennaro, M.: Numerical methods for delay differential equations. Oxford University Press, New York (2003). doi:10.1093/acprof:oso/9780198506546.001.0001 · Zbl 1038.65058 |
| [9] | Bellman, R.; Cooke, KL, Differential-difference equations, 1963, New York-London: Academic Press, New York-London · Zbl 0105.06402 |
| [10] | Biot, MA, General theory of three-dimensional consolidation, J. Appl. Phys., 12, 2, 155-164, 1941 · JFM 67.0837.01 · doi:10.1063/1.1712886 |
| [11] | Biot, MA, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys., 27, 240-253, 1956 · Zbl 0071.41204 · doi:10.1063/1.1722351 |
| [12] | Chaabane, N.; Rivière, B., A sequential discontinuous Galerkin method for the coupling of flow and geomechanics, J. Sci. Comput., 74, 1, 375-395, 2018 · Zbl 1404.65162 · doi:10.1007/s10915-017-0443-6 |
| [13] | Chaabane, N.; Rivière, B., A splitting-based finite element method for the Biot poroelasticity system, Comput. Math. Appl., 75, 7, 2328-2337, 2018 · Zbl 1409.76057 · doi:10.1016/j.camwa.2017.12.009 |
| [14] | Ciarlet, PG, Mathematical elasticity, 1988, Amsterdam: North-Holland, Amsterdam · Zbl 0648.73014 |
| [15] | Crouzeix, M., Une méthode multipas implicite-explicite pour l’approximation des équations d’évolution paraboliques, Numer. Math., 35, 3, 257-276, 1980 · Zbl 0419.65057 · doi:10.1007/BF01396412 |
| [16] | Detournay, E., Cheng, A.H.D.: Fundamentals of poroelasticity. In: Analysis and design methods, pp. 113-171. Elsevier (1993) |
| [17] | Ern, A.; Meunier, S., A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems, ESAIM: Math. Model. Numer. Anal., 43, 2, 353-375, 2009 · Zbl 1166.76036 · doi:10.1051/m2an:2008048 |
| [18] | Frank, J.; Hundsdorfer, W.; Verwer, J., On the stability of implicit-explicit linear multistep methods, Appl. Numer. Math., 25, 2, 193-205, 1997 · Zbl 0887.65094 · doi:10.1016/S0168-9274(97)00059-7 |
| [19] | Fu, G., A high-order HDG method for the Biot’s consolidation model, Comput. Math. Appl., 77, 1, 237-252, 2019 · Zbl 1442.65257 · doi:10.1016/j.camwa.2018.09.029 |
| [20] | Gu, K.; Kharitonov, VL; Chen, J., Stability of Time-Delay Systems, 2003, Boston: Birkhäuser, Boston · Zbl 1039.34067 · doi:10.1007/978-1-4612-0039-0 |
| [21] | Hairer, E., Wanner, G.: Solving ordinary differential equations. II, Springer Series in Computational Mathematics, vol. 14, second edn. Springer-Verlag, Berlin (1996). doi:10.1007/978-3-642-05221-7. Stiff and differential-algebraic problems · Zbl 0859.65067 |
| [22] | Kim, J.; Tchelepi, HA; Juanes, R., Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits, Comput. Methods Appl. Mech. Engrg., 200, 13-16, 1591-1606, 2011 · Zbl 1228.74101 · doi:10.1016/j.cma.2010.12.022 |
| [23] | Kunkel, P., Mehrmann, V.: Differential-algebraic equations. Analysis and numerical solution. European mathematical society, Zürich (2006). doi:10.4171/017 · Zbl 1095.34004 |
| [24] | Lee, JJ; Mardal, KA; Winther, R., Parameter-robust discretization and preconditioning of Biot’s consolidation model, SIAM J. Sci. Comput., 39, 1, A1-A24, 2017 · Zbl 1381.76183 · doi:10.1137/15M1029473 |
| [25] | Mikelić, A.; Wheeler, MF, Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 17, 3, 455-461, 2013 · Zbl 1392.35235 · doi:10.1007/s10596-012-9318-y |
| [26] | Mujahid, A.: Monolithic, non-iterative and iterative time discretization methods for linear coupled elliptic-parabolic systems. GAMM Archive for students 4(1) (2022). doi:10.14464/gammas.v4i1.500 |
| [27] | Showalter, RE, Diffusion in poro-elastic media, J. Math. Anal. Appl., 251, 1, 310-340, 2000 · Zbl 0979.74018 · doi:10.1006/jmaa.2000.7048 |
| [28] | Storvik, E.; Both, JW; Kumar, K.; Nordbotten, JM; Radu, FA, On the optimization of the fixed-stress splitting for Biot’s equations, Int. J. Numer. Meth. Eng., 120, 2, 179-194, 2019 · Zbl 07859732 · doi:10.1002/nme.6130 |
| [29] | Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, second edn. Springer series in computational mathematics. Springer Berlin, Heidelberg (2006). doi:10.1007/3-540-33122-0 · Zbl 1105.65102 |
| [30] | Trenn, S., Unger, B.: Delay regularity of differential-algebraic equations. In: 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France, pp. 989-994 (2019). doi:10.1109/CDC40024.2019.9030146 |
| [31] | Unger, B.: Discontinuity propagation in delay differential-algebraic equations. Electron. J. Linear Algebr. 34, 582-601 (2018). doi:10.13001/1081-3810.3759 · Zbl 1417.34188 |
| [32] | Wheeler, MF; Gai, X., Iteratively coupled mixed and Galerkin finite element methods for poro-elasticity, Numer. Meth. Part. D. E., 23, 4, 785-797, 2007 · Zbl 1115.74054 · doi:10.1002/num.20258 |
| [33] | Zeidler, E., Nonlinear Functional Analysis and its Applications IIa: Linear Monotone Operators, 1990, New York: Springer-Verlag, New York · Zbl 0684.47029 · doi:10.1007/978-1-4612-0981-2 |
| [34] | Zoback, MD, Reservoir Geomechanics, 2010, Cambridge: Cambridge University Press, Cambridge · Zbl 1270.62001 |
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