nach oben

Journal of Scientific Computing

Erschienen in:

09.02.2018

Convergence of the MAC Scheme for the Stokes/Darcy Coupling Problem

verfasst von: Ming-Cheng Shiue, Kian Chuan Ong, Ming-Chih Lai

Erschienen in: Journal of Scientific Computing | Ausgabe 2/2018

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper, we extend the MAC scheme for Stokes problem to the Stokes/Darcy coupling problem. The interface conditions between two separate regions are discretized and well-incorporated into the MAC grid setting. We first perform the stability analysis of the scheme for the velocity in both Stokes and Darcy regions and establish the stability for the pressure in both regions by considering an analogue of discrete divergence problem. Following the similar analysis on stability, we perform the error estimates for the velocity and the pressure in both regions. The theoretical results show the first-order convergence of the scheme in discrete \(L^2\) norms for both velocity and the pressure in both regions. Moreover, in fluid region, the first-order convergence for the x-derivative of velocity component u and the y-derivative of velocity component v is also obtained in discrete \(L^2\) norms. However, numerical tests show one order better for the velocity in Stokes region and the pressure in Darcy region.

Vorheriger Artikel A Concurrent Global–Local Numerical Method for Multiscale PDEs

Nächster Artikel Unconditional Convergence in Maximum-Norm of a Second-Order Linearized Scheme for a Time-Fractional Burgers-Type Equation

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Arbogast, T., Brunson, D.S.: A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium. Comput. Geosci. 11, 207–218 (2007)MathSciNetCrossRefMATH

Tlupova, S., Cortez, R.: Boundary integral solutions of coupled Stokes and Darcy flows. J. Comput. Phys. 228, 158–179 (2009)MathSciNetCrossRefMATH

Chen, Q.: Stable and convergent approximation of two-dimensional vector fields on unstructured meshes. J. Comput. Appl. Math. 307, 284–306 (2016)MathSciNetCrossRefMATH

Chou, S.H.: Analysis and convergence of a covolume method for the generalized Stokes problem. Math. Comp. 66, 85–104 (1997)MathSciNetCrossRefMATH

Cai, M., Mu, M., Xu, J.: Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications. J. Comput. Appl. Math. 233, 346–355 (2009)MathSciNetCrossRefMATH

Cao, Y., Gunzburger, M., Hu, X., Hua, F., Wang, X., Zhao, W.: Finite element approximations for Stokes–Darcy flow with Beavers–Joseph interface conditions. SIAM J. Numer. Anal. 47, 4239–4256 (2010)MathSciNetCrossRefMATH

Cao, Y., Gunzburger, M., Hu, X., Wang, X.: Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes–Darcy systems. Math. Comput. 83, 1617–1644 (2014)MathSciNetCrossRefMATH

Camano, J., Gatica, G.N., Oyarza, R., Ruiz-Baier, R., Venegas, P.: New fully-mixed finite element methods for the Stokes–Darcy coupling. Comput. Methods Appl. Mech. Eng. 295, 362–395 (2015)MathSciNetCrossRef

Chidyagwai, P., Ladenheim, S., Szyld, D.: Constraint preconditioning for the coupled Stokes–Darcy system. SIAM J. Sci. Comput. 38, A668–A690 (2016)MathSciNetCrossRefMATH

10.

Chidyagwai, P., Riviere, B.: Numerical modelling of coupled surface and subsurface flow systems. Adv. Water Res. 33, 92–105 (2010)CrossRef

11.

Chen, W., Gunzburger, M., Sun, D., Wang, X.: An efficient and long-time accurate third-order algorithm for the Stokes–Darcy system. Numer. Math. 134, 857–879 (2016)MathSciNetCrossRefMATH

12.

Discacciati, M., Quarteroni, A.: Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations. Comput. Visual Sci. 6, 93–103 (2004)MathSciNetCrossRefMATH

13.

Discacciati, M., Quarteroni, A.: Navier–Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Complut. 22, 315–426 (2009)MathSciNetCrossRefMATH

14.

Gatica, G., Oyarzua, R., Sayas, F.: Analysis of fully-mixed finite element methods for the Stokes–Darcy coupled problem. Math. Comput. 80, 1911–1948 (2011)MathSciNetCrossRefMATH

15.

Girault, V., Vassilev, D., Yotov, I.: Mortar multiscale finite element methods for Stokes–Darcy flows. Numer. Math. 127, 93–165 (2014)MathSciNetCrossRefMATH

16.

Harlow, F.H., Welsh, J.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with a free interface. Phys. Fluids 8, 2181–2189 (1965)CrossRef

17.

Han, H., Wu, X.: A new mixed finite element formulation and the MAC method for the Stokes equations. SIAM J. Numer. Anal. 35, 650–571 (1998)MathSciNetCrossRef

18.

Hessari, P.: Pseudospectral least squares method for Stokes–Darcy equations. SIAM J. Numer. Anal. 53, 1195–1213 (2015)MathSciNetCrossRefMATH

19.

Kanschat, G., Rivire, B.: A strongly conservative finite element method for the coupling of Stokes and Darcy flow. J. Comput. Phys. 229, 5933–5943 (2010)MathSciNetCrossRefMATH

20.

Layton, W.J., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40(6), 2195–2218 (2003)MathSciNetCrossRefMATH

21.

Li, Z.: An augmented Cartesian grid method for Stokes–Darcy fluid–structure interactions. Int. J. Numer. Meth. Eng. 106, 556–575 (2016)MathSciNetCrossRefMATH

22.

Li, J., Sun, S.: The superconvergence phenomenon and proof of the MAC scheme for the Stokes equations on non-uniform rectangular meshes. J. Sci. Comput. 65, 341–362 (2015)MathSciNetCrossRefMATH

23.

Mu, M., Xu, J.: A two-grid method of a mixed Stokes–Darcy model for coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 45, 1801–1813 (2007)MathSciNetCrossRefMATH

24.

Nicolaides, R.A.: Analysis and convergence of the MAC scheme I. The linear problem. SIAM J. Numer. Anal. 29, 1579–1591 (1992)MathSciNetCrossRefMATH

25.

Rui, H., Li, X.: Stability and superconvergence of MAC scheme for Stokes equations on nonuniform grids. SIAM J. Numer. Anal. 55(3), 1135–1158 (2017)MathSciNetCrossRefMATH

26.

Rui, H., Liu, W.: A two-grid block-centered finite difference method MAC scheme forDarcy–Forchheimer flow in porous media. SIAM J. Numer. Anal. 53(4), 1941–1962 (2015)MathSciNetCrossRefMATH

27.

Rui, H., Pan, H.: A block-centered finite difference method for the Darcy–Forchheimer. SIAM J. Numer. Anal. 50(5), 2612–2631 (2012)MathSciNetCrossRefMATH

28.

Rui, H., Pan, H.: A block-centered finite difference method for slightly compressible Darcy–Forchheimer flow in porous media. J. Sci. Comput. 73(1), 70–92 (2017)MathSciNetCrossRefMATH

29.

Shin, D., Strikwerda, J.C.: Inf-sup conditions for finite-difference approximations of the Stokes equations. J. Aust. Math. Soc. Ser. B 39, 121–134 (1997)MathSciNetCrossRefMATH

30.

Wang, W., Xu, C.: Spectral methods based on new formulations for coupled Stokes and Darcy equations. J. Comput. Phys. 257, 126–142 (2014)MathSciNetCrossRefMATH

Titel: Convergence of the MAC Scheme for the Stokes/Darcy Coupling Problem
verfasst von: Ming-Cheng Shiue
Kian Chuan Ong
Ming-Chih Lai
Publikationsdatum: 09.02.2018
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 2/2018
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0660-7

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 2/2018

An HDG Method with Orthogonal Projections in Facet Integrals

A Concurrent Global–Local Numerical Method for Multiscale PDEs

Strong Stability Preserving General Linear Methods with Runge–Kutta Stability

Unconditional Convergence in Maximum-Norm of a Second-Order Linearized Scheme for a Time-Fractional Burgers-Type Equation

Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method

Galerkin Methods for Stationary Radiative Transfer Equations with Uncertain Coefficients