Top

Journal of Scientific Computing

Published in:

09-10-2017

Conservative Local Discontinuous Galerkin Method for Compressible Miscible Displacements in Porous Media

Authors: Fan Yu, Hui Guo, Nattaporn Chuenjarern, Yang Yang

Published in: Journal of Scientific Computing | Issue 2-3/2017

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In Guo et al. (Appl Math Comput 259:88–105, 2015), a nonconservative local discontinuous Galerkin (LDG) method for both flow and transport equations was introduced for the one-dimensional coupled system of compressible miscible displacement problem. In this paper, we will continue our effort and develop a conservative LDG method for the problem in two space dimensions. Optimal error estimates in \(L^{\infty }(0, T; L^{2})\) norm for not only the solution itself but also the auxiliary variables will be derived. The main difficulty is how to treat the inter-element discontinuities of two independent solution variables (one from the flow equation and the other from the transport equation) at cell interfaces. Numerical experiments will be given to confirm the accuracy and efficiency of the scheme.

previous article Discontinuous Galerkin Methods for Relativistic Vlasov–Maxwell System

next article Fourier Type Super Convergence Study on DDGIC and Symmetric DDG Methods

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

inform now

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!

inform now

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!

inform now

Available only for authorised users

Bartels, S., Jensen, M., Müller, R.: Discontinuous Galerkin finite element convergence for incompressible miscible displacement problem of low regularity. SIAM J. Numer. Analy. 47, 3720–3743 (2009)CrossRefMATHMathSciNet

Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131, 267–279 (1997)CrossRefMATHMathSciNet

Bell, J., Dawson, C.N., Shubin, G.R.: An unsplit high-order Godunov scheme for scalar conservation laws in two dimensions. J. Comput. Phys. 74, 1–24 (1988)CrossRefMATH

Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: Superconvergence of the local discontinuous Galerkin method for elliptic problems on cartesian grids. SIAM J. Numer. Anal. 39, 264–285 (2001)CrossRefMATHMathSciNet

Ciarlet, P.: The Finite Element Method for Elliptic Problem. North Holland, Amsterdam (1975)

Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)CrossRefMATHMathSciNet

Cui, M.: Analysis of a semidiscrete discontinuous Galerkin scheme for compressible miscible displacement problem. J. Comput. Appl. Math. 214, 617–636 (2008)CrossRefMATHMathSciNet

Douglas Jr., J., Roberts, J.E.: Numerical methods for a model for compressible miscible displacement in porous media. Math. Comput. 41, 441–459 (1983)CrossRefMATHMathSciNet

Douglas Jr., J., Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristic with finite element of finite difference procedures. SIAM J. Numer. Anal. 19, 871–885 (1982)CrossRefMATHMathSciNet

10.

Douglas Jr., J., Ewing, R.E., Wheeler, M.F.: A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Anal. Numér. 17, 249–256 (1983)CrossRefMATHMathSciNet

11.

Douglas Jr., J., Ewing, R.E., Wheeler, M.F.: The approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO Anal. Numér. 17, 17–33 (1983)CrossRefMATHMathSciNet

12.

Ewing, R.E., Russell, T.F., Wheeler, M.F.: Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput. Methods Appl. Mech. Eng. 47, 73–92 (1984)CrossRefMATHMathSciNet

13.

Gelfand, I.M.: Some questions of analysis and differential equations. Am. Math. Soc. Transl. 26, 201–219 (1963)MathSciNet

14.

Guo, H., Zhang, Q., Wang, J.: Error analysis of the semi-discrete local discontinuous Galerkin method for compressible miscible displacement problem in porous media. Appl. Math. Comput. 259, 88–105 (2015)MathSciNet

15.

Guo, H., Yu, F., Yang, Y.: Local discontinuous Galerkin method for incompressible miscible displacement problem in porous media. J. Sci. Comput. 71, 615–633 (2017)CrossRefMathSciNet

16.

Hurd, A.E., Sattinger, D.H.: Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients. Trans. Am. Math. Soc. 132, 159–174 (1968)CrossRefMATHMathSciNet

17.

Ji, L., Xu, Y.: Optimal error estimates of the local discontinuous Galerkin method for Willmore flow of graphs on Cartesian meshes. Int. J. Numer. Anal. Model. 8, 252–283 (2011)MATHMathSciNet

18.

Johnson, C.: Streamline Diffusion Methods for Problems in Fluid Mechanics. Finite Element in Fluids VI. Wiley, New York (1986)

19.

Li, X., Rui, H., Xu, W.: A new MCC–MFE method for compressible miscible displacement in porous media. J. Comput. Appl. Math. 302, 139–156 (2016)CrossRefMATHMathSciNet

20.

Rivière, B.: Discontinuous Galerkin finite element methods for solving the miscible displacement problem in porous media. In: Ph.D. thesis, The University of Texas at Austin (2000)

21.

Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)CrossRefMATHMathSciNet

22.

Sun, S., Rivière, B., Wheeler, M.: A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media. In: Chan, T., et al. (eds.) Recent Progress in Computational and Applied PDEs, pp. 323–351. Kluwer Academic Publishers, Plenum Press, Dordrecht, NewYork (2002)CrossRef

23.

Sun, S., Wheeler, M.: Discontinuous Galerkin methods for coupled flow and reactive transport problems. Appl. Numer. Math. 52, 273–298 (2005)CrossRefMATHMathSciNet

24.

Wang, H., Shu, C.-W., Zhang, Q.: Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems. SIAM J. Numer. Anal. 53, 206–227 (2015)CrossRefMATHMathSciNet

25.

Wang, H., Shu, C.-W., Zhang, Q.: Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for nonlinear convection-diffusion problems. Appl. Math. Comput. 272, 237–258 (2016)MathSciNet

26.

Wang, H., Wang, S., Zhang, Q., Shu, C.-W.: Local discontinuous Galerkin methods with implicit-explicit time marching for multi-dimensional convectiondiffusion problems. ESAIM: M2AN 50, 1083–1105 (2016)CrossRefMATH

27.

Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for nonlinear Schrodinger equations. J. Comput. Phys. 205, 72–97 (2005)CrossRefMATHMathSciNet

28.

Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations. Comput. Methods Appl. Mech. Eng. 195, 3430–3447 (2006)CrossRefMATHMathSciNet

29.

Yan, J., Shu, C.-W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40, 769–791 (2002)CrossRefMATHMathSciNet

30.

Yang, Y., Shu, C.-W.: Analysis of optimal superconvergence of local discontinuous Galerkin method for one-dimensional linear parabolic equations. J. Comput. Math. 33, 323–340 (2015)CrossRefMathSciNet

31.

Yuan, Y.: Characteristic finite element methods for positive semidefinite problem of two phase miscible flow in three dimensions. Chin. Sci. Bull. 22, 2027–2032 (1996)

32.

Zhang, J., Yang, D., Shen, S., Zhu, J.: A new MMOCAA-MFE method for compressible miscible displacement in porous media. Appl. Numer. Math. 80, 65–80 (2014)CrossRefMATHMathSciNet

Title: Conservative Local Discontinuous Galerkin Method for Compressible Miscible Displacements in Porous Media
Authors: Fan Yu
Hui Guo
Nattaporn Chuenjarern
Yang Yang
Publication date: 09-10-2017
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2-3/2017
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0571-z

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 2-3/2017

An Augmented Method for 4th Order PDEs with Discontinuous Coefficients

Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

Weighted Nonlocal Laplacian on Interpolation from Sparse Data

Local Discontinuous Galerkin Methods for the Khokhlov–Zabolotskaya–Kuznetzov Equation

Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Schrödinger Equations

A Hybrid Method to Solve Shallow Water Flows with Horizontal Density Gradients

Premium Partner