Abstract
In this paper, we present Eulerian-Lagrangian localized adjoint methods (ELLAM) to solve convection-diffusion-reaction equations governing contaminant transport in groundwater flowing through an adsorbing porous medium. These ELLAM schemes can treat various combinations of boundary conditions and conserve mass. Numerical results are presented to demonstrate the strong potential of ELLAM schemes.
This research was supported in part by ONR Contract No. 0014-88-K-0370, by NSF Grant No. DMS-8922865, by funding from the Institute for Scientific Computation at Texas A&M University, by DOE, DE-ACO5-840R21400, Martin Marietta, Subcontracts, SK965C and SK966V, by DEPSCoR 94 Grant, and by funding from the Norwegian Research Council for Science and Humanities.
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References
M.B. ALLEN III, G.A. BEHIE, AND J.A. TRANGENSTEIN, Multiphase flow in porous media, Lecture Notes in Engineering, Springer-Verlag, Berlin (1988).
J.W. BARRETT AND K.W. MORTON, Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems, Comp. Meth. Appl. Mech. Engrg., 45 (1984), pp. 97–122.
A. BROOKS AND T.J.R. HUGHES, Streamline upwind Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comp. Meth. Appl. Mech. Engrg., 32 (1982), pp. 199–259.
M.A. CELIA, I. HERRERA, E.T. BOULOUTAS, AND J.S. KINDRED, A new numerical approach for the advective-diffusive transport equation, Numerical Methods for PDE’s, 5 (1989), pp. 203–226.
M.A. CELIA, T.F. RUSSELL, I. HERRERA, AND R.E. EWING, An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation, Advances in Water Resources, 13 (1990), pp. 187–206.
M.A. CELIA AND S. ZISMAN, An Eulerian-Lagrangian localized adjoint method for reactive transport in groundwater, Computational Methods in Subsurface Hydrology, Proceedings of the Eighth International Conference on Computational Methods in Water Resources, Venice, Italy (1990), pp. 383–392.
H.K. DAHLE, M.S. ESPEDAL, AND R.E. EWING, Characteristic Petrov-Galerkin subdomain methods for convection diffusion problems (M.F. Wheeler, ed.), IMA Volume 11, Numerical Simulation in Oil Recovery, Springer-Verlag, Berlin (1988), pp. 77–88.
H.K. DAHLE, M.S. ESPEDAL, R.E. EWING, AND O. SEVAREID, Characteristic adaptive sub-domain methods for reservoir flow problems, Numerical Methods for PDE’s, 6, (1990), pp. 279–309.
H.K. DAHLE, R.E. EWING, AND T.F. RUSSELL, Eulerian-Lagrangian localized adjoint methods for a nonlinear convection-diffusion equation. Comp. Meth. Appl. Mech. Engrg. Vol. 122, Nos. 3–4, (1995), 223–250.
C.N. DAWSON, T.F. RUSSELL, AND M.F. WHEELER, Some improved error estimates for the modified method of characteristics, SIAM J. Numer. Anal., 26 (1989), pp. 1487–1752.
L. DEMKOWITZ AND J.T. ODEN, An adaptive characteristic Petrov-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in two space variables, Comp. Meth. Appl. Mech. Engrg., 55 (1986), pp. 63–87.
L. DEMKOWITZ AND J.T. ODEN, An adaptive characteristic Petrov-Galerkin method for convection-dominated linear and nonlinear parabolic problems in one space variable, J. Comp. Phys. 67 (1986), pp. 188–213.
L. DEMKOWITZ AND J.T. ODEN, An adaptive characteristic Petrov-Galerkin method for convection-dominated linear and nonlinear parabolic problems in one space variable, J. Comp. Phys. 67 (1986), pp. 188–213.
M.S. ESPEDAL AND R.E. EWING, Characteristic Petrov-Galerkin subdomain methods for two-phase immiscible flow, Comp. Meth. Appl. Mech. Engrg., 64 (1987), pp. 113–135.
R.E. EWING, ed., Research Frontiers in Applied Mathematics, Vol. 1, SIAM, Philadelphia (1983).
R.E. EWING, Operator splitting and Eulerian-Lagrangian localized adjoint methods for multiphase flow. Whiteman, J. (ed.), The Mathematics of Finite Elements and Applications VII (MAFELAP 1990). Academic Press, Inc., San Diego, (1990), 215–232.
R.E. EWING AND H. WANG, Eulerian-Lagrangian localized adjoint methods for linear advection or advection-reaction equations and their convergence analysis. Computational Mechanics, 12 (1993), 97–121.
I. HERRERA, The algebraic theory approach for ordinary differential equations: highly accurate finite differences, Numerical Methods for PDE’s, 3 (1987), pp. 199–218.
T.J.R. HUGHES AND A. BROOKS, A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions. Applications to the streamline-upwind procedure (R.H. Gallagher, ed.), Finite Elements in Flu¬ids, 4, Wiley, New York (1982).
C. JOHNSON, Numerical solutions of partial differential equations by the finite element method, Cambridge University Press, Cambridge (1987).
T.F. RUSSELL, Time-stepping along characteristics with incomplete iteration for a
Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal., 22 (1985), pp. 970–1013.
T.F. RUSSELL, Eulerian-Lagrangian localized adjoint methods for advection-dominated problems, Numerical Analysis 1989, Proceedings of the 13th Dundee Conference on Numerical Analysis (D.F. Griffiths and G.A. Watson, eds.), Pitman Research Notes in Mathematics Series, 228, Longman Scientific & Technical, Harlow, U.K. (1989), pp. 206–228.
T.F. RUSSELL AND R.V. TRUJILLO, Eulerian-Lagrangian localized adjoint methods with variable coefficients in multiple dimensions, Computational Methods in Surface Hydrology, Proceedings of the Eighth International Conference on Computational Methods in Water Resources, Venice, Italy (1990), pp. 357–363.
T.F. RUSSELL AND M.F. WHEELER, Finite element and finite difference methods for continuous flows in porous media (R.E. Ewing, ed.), Mathematics of Reservoir Simulation, Frontiers in Applied Math, Philadelphia, Pennsylvania (1983), pp. 35–106.
E. VAROGLU AND W.D.L. FINN, Finite elements incorporating characteristics for one-dimensional diffusion-convection equation, J. Comp. Phys., 34 (1980), pp. 371–389.
H. WANG, R.E. EWING, AND T.F. RUSSELL, Eulerian-Lagrangian localized methods for convection-diffusion equations and their convergence analysis. IMA J. Numer. Anal., (submitted).
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Ewing, R.E., Wang, H. (1996). Eulerian-Lagrangian Localized Adjoint Methods for Reactive Transport in Groundwater. In: Wheeler, M.F. (eds) Environmental Studies. The IMA Volumes in Mathematics and its Applications, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8492-2_6
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