×

A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations. (English) Zbl 0956.76050

Summary: We develop a family of Eulerian-Lagrangian localized adjoint methods (ELLAM) for the solution of initial-boundary value problems for first-order advection-reaction equations on general multidimensional domains. Different tracking algorithms, including the Euler and Runge-Kutta algorithms, are used. The derived schemes, which are fully mass conservative, naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary conditions. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices, which can be efficiently solved by the conjugate gradient method in an optimal order number of iterations without any preconditioning needed.
Numerical result are presented to compare the performance of the ELLAM schemes with many well studied and widely used methods, including upwind finite difference methods, Galerkin and the Petrov-Galerkin finite element methods with backward-Euler or Crank-Nicolson temporal discretization, streamline diffusion finite element methods, the monotonic upstream-centered scheme for conservation laws (MUSCL), and the minmod scheme.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76R99 Diffusion and convection
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allen, M. B.; Behie, G. A.; Trangenstein, J. A., Multiphase Flow in Porous Media, 34 (1988)
[2] Aziz, H.; Settari, A., Petroleum Reservoir Simulation (1979)
[3] Baptista, A. M., Solution of Advection-Dominated Transport by Eulerian-Lagrangian Methods Using the Backwards Method of Characteristics (1987)
[4] Barrett, J. W.; Morton, K. W., Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems, Comput. Methods Appl. Mech. Eng., 45, 97 (1984) · Zbl 0562.76086
[5] Bear, J., Hydraulics of Groundwater (1979)
[6] Binning, P. J.; Celia, M. A., A finite volume Eulerian-Lagrangian localized adjoint method for solution of the contaminant transport equations in two-dimensional multiphase flow systems, Water Resources Res., 32, 103 (1996)
[7] Borden, R. C.; Bedient, P. B., Transport of dissolved hydrocarbons influenced by oxygen-limited biodegradation. 1. Theoretical development, Water Resources Res., 22, 1973 (1986)
[8] Bouloutas, E. T.; Celia, M. A., An improved cubic Petrov-Galerkin method for simulation of transient advection-diffusion processes in rectangularly decomposable domains, Comput. Methods Appl. Mech. Eng., 91, 289 (1991) · Zbl 0825.76609
[9] Brooks, A.; Hughes, T. J.R, Streamline upwind Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 32, 199 (1982) · Zbl 0497.76041
[10] Buckley, S. E.; Leverret, M. C., Mechanism of fluid displacement in sands, Trans. AIME, 146, 107 (1942)
[11] Celia, M. A.; Herrera, I.; Bouloutas, E. T.; Kindred, J. S., A new numerical approach for the advective-diffusive transport equation, Numer. Methods PDEs, 5, 203 (1989) · Zbl 0678.65083
[12] Celia, M. A.; Russell, T. F.; Herrera, I.; Ewing, R. E., An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation, Adv. Water Resources, 13, 187 (1990)
[13] Chavent, G.; Jaffré, J., Mathematical Models and Finite Elements for Reservoir Simulation (1986) · Zbl 0603.76101
[14] Christie, I.; Griffiths, D. F.; Mitchell, A. R.; Zienkiewicz, O. C., Finite element methods for second order differential equations with significant first derivatives, Int. J. Numer. Eng., 10, 1389 (1976) · Zbl 0342.65065
[15] Colella, P., A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. Sci. Stat. Comp., 6, 104 (1985) · Zbl 0562.76072
[16] Courant, R.; Isaacson, E.; Rees, M., On the solution of nonlinear hyperbolic differential equations by finite differences, Comm. Pure Appl. Math., 5, 243 (1952) · Zbl 0047.11704
[17] Cox, R. A.; Nishikawa, T., A new total variation diminishing scheme for the solution of advective-dominant solute transport, Water Resources Res., 27, 2645 (1991)
[18] Crandall, M. G.; Majda, A., Monotone difference approximations for scalar conservation laws, Math. Comp., 34, 1 (1980) · Zbl 0423.65052
[19] Dahle, H. K.; Espedal, M. S.; Ewing, R. E.; Sæ vareid, O., Characteristic adaptive sub-domain methods for reservoir flow problems, Numer. Methods PDEs, 6, 279 (1990) · Zbl 0707.76093
[20] Dahle, H. K.; Ewing, R. E.; Russell, T. F., Eulerian-Lagrangian localized adjoint methods for a nonlinear convection-diffusion equation, Comput. Methods Appl. Mech. Eng., 122, 223 (1995) · Zbl 0851.76058
[21] Douglas, J.; Russell, T. F., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19, 871 (1982) · Zbl 0492.65051
[22] Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., 25, 294 (1988) · Zbl 0642.76088
[23] Espedal, M. S.; Ewing, R. E., Characteristic Petrov-Galerkin subdomain methods for two-phase immiscible flow, Comput. Methods Appl. Mech. Eng., 64, 113 (1987) · Zbl 0607.76103
[24] Ewing, R. E., The Mathematics of Reservoir Simulation, 1 (1984) · Zbl 0535.76120
[25] R. E. Ewing, Operator splitting and Eulerian-Lagrangian localized adjoint methods for multiphase flow, in, The Mathematics of Finite Elements and Applications, edited by, Whiteman, Academic Press, San Diego, CA, 1991, Vol, VII, p, 215.; R. E. Ewing, Operator splitting and Eulerian-Lagrangian localized adjoint methods for multiphase flow, in, The Mathematics of Finite Elements and Applications, edited by, Whiteman, Academic Press, San Diego, CA, 1991, Vol, VII, p, 215.
[26] Ewing, R. E.; Heinemann, R. F., Mixed finite element approximation of phase velocities in compositional reservoir simulation, Comput. Methods Appl. Mech. Eng., 47, 161 (1984) · Zbl 0545.76127
[27] Ewing, R. E.; Russell, T. F.; Wheeler, M. F., Simulation of miscible displacement using mixed methods and a modified method of characteristics, SPE, 12241, 71 (1983)
[28] 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 (1984) · Zbl 0545.76131
[29] Ewing, R. E.; Wang, H., Eulerian-Lagrangian localized adjoint methods for linear advection equations, Computational Mechanics 1991, 245 (1991)
[30] Ewing, R. E.; Wang, H., An optimal-order error estimate to Eulerian-Lagrangian localized adjoint method for variable-coefficient advection-reaction problems, SIAM Numer. Anal., 33, 318 (1996) · Zbl 0860.65086
[31] R. E. Ewing, H. Wang, M. A. Celia, and, R. Sharpley, A three-dimensional finite element simulation of nuclear waste contamination transport in porous media, in, Computer Methods and Advances in Geomechanics, edited by, Siriwardane and Zaman, Balkema, Rotterdam, 1995, Vol, 9, p, 2673.; R. E. Ewing, H. Wang, M. A. Celia, and, R. Sharpley, A three-dimensional finite element simulation of nuclear waste contamination transport in porous media, in, Computer Methods and Advances in Geomechanics, edited by, Siriwardane and Zaman, Balkema, Rotterdam, 1995, Vol, 9, p, 2673.
[32] R. E. Ewing, H. Wang, and, R. C. Sharpley, Eulerian-Lagrangian localized adjoint methods for transport of nuclear-waste contamination in porous media, in, Computational Methods in Water Resources X, Vol, 1, edited by, Peters, et al.; R. E. Ewing, H. Wang, and, R. C. Sharpley, Eulerian-Lagrangian localized adjoint methods for transport of nuclear-waste contamination in porous media, in, Computational Methods in Water Resources X, Vol, 1, edited by, Peters, et al.
[33] Ewing, R. E.; Yuan, Y.; Li, G., Time stepping along characteristics for a mixed finite element approximation for compressible flow of contamination by nuclear waste in porous media, SIAM J. Numer. Anal., 26, 1513 (1989) · Zbl 0681.76096
[34] Finlayson, B. A., Numerical Methods for Problems with Moving Fronts (1992)
[35] Garder, A. O.; Peaceman, D. W.; Pozzi, A. L., Numerical calculations of multidimensional miscible displacement by the method of characteristics, Soc. Pet. Eng. J., 4, 26 (1964)
[36] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order accurate essentially nonoscillatory schemes, III, J. Comput. Phys., 71, 231 (1987) · Zbl 0652.65067
[37] Healy, R. W.; Russell, T. F., A finite-volume Eulerian-Lagrangian localized adjoint method for solution of the advection-dispersion equation, Water Resources Res., 29, 2399 (1993)
[38] Herrera, I.; Ewing, R. E.; Celia, M. A.; Russell, T. F., Eulerian-Lagrangian localized adjoint methods: The theoretical framework, Numer. Methods PDEs, 9, 431 (1993) · Zbl 0784.65071
[39] Hillel, D., Fundamentals of Soil Physics (1980)
[40] T. J. R. Hughes, and, A. N. Brooks, A multidimensional upwinding scheme with no crosswind diffusion, in, Finite Element Methods for Convection Dominated Flows, edited by, Hughes, ASME, New York, 1979, Vol, 34.; T. J. R. Hughes, and, A. N. Brooks, A multidimensional upwinding scheme with no crosswind diffusion, in, Finite Element Methods for Convection Dominated Flows, edited by, Hughes, ASME, New York, 1979, Vol, 34.
[41] Hughes, T. J.R; Mallet, M., A new finite element formulation for computational fluid dynamics. III. The general streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Eng., 58, 305 (1986) · Zbl 0622.76075
[42] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Eng., 45, 285 (1984) · Zbl 0526.76087
[43] Johnson, C.; Pitkäranta, J., An analysis of discontinuous Galerkin methods for a scalar hyperbolic equation, Math. Comp., 46, 1 (1986) · Zbl 0618.65105
[44] Johnson, C.; Szepessy, A., Adaptive finite element methods for conservation laws based on a posteriori error estimates, Comm. Pure Appl. Math., 98, 199 (1995) · Zbl 0826.65088
[45] Johnson, C.; Szepessy, A.; Hansbo, P., On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp., 54, 107 (1990) · Zbl 0685.65086
[46] Kindred, J. S.; Celia, M. A., Contaminant transport and biodegradation. II. Conceptual model and test simulations, Water Resources Res., 25, 1149 (1989)
[47] Lax, P. D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1973) · Zbl 0268.35062
[48] LeVeque, R. J., Numerical Methods for Conservation Laws (1992) · Zbl 0847.65053
[49] Morton, K. W.; Priestley, A.; Süli, E., Stability of the Lagrangian-Galerkin method with nonexact integration, RAIRO Model. Math. Anal. Numer., 22, 123 (1988) · Zbl 0661.65114
[50] Neuman, S. P., An Eulerian-Lagrangian numerical scheme for the dispersion-convection equation using conjugate space-time grids, J. Comput. Phys., 41, 270 (1981) · Zbl 0484.76095
[51] Peaceman, D. W., Fundamentals of Numerical Reservoir Simulation (1977) · Zbl 0204.28001
[52] Pinder, G. F.; Cooper, H. H., A numerical technique for calculating the transient position of the saltwater front, Water Resources Res., 6, 875 (1970)
[53] Pironneau, O., On the transport-diffusion algorithm and its application to the Navier-Stokes equations, Numer. Math., 38, 309 (1982) · Zbl 0505.76100
[54] Pollock, D. W., Semianalytical computation of path lines for finite-difference models, Ground Water, 26, 743 (1988)
[55] Pudykiewicz, J.; Staniforth, A., Some properties and comparative performance of the semi-Lagrangian method of robert in the solution of the advection-diffusion equation, Atmosphere-Ocean., 22, 283 (1984)
[56] Qin, G.; Wang, H.; Ewing, R. E.; Espedal, M. S., Efficient numerical solution techniques for compositional model, Iterative Methods in Linear Algebra, II, 427 (1996)
[57] Reeves, M.; Cranwell, R. M., User’s Manual for the Sandia Waste-Isolation Flow and Transport Model (SWIFT) Release 4.81 (1981)
[58] Reeves, M.; Ward, D. S.; Johns, N. D.; Cranwell, R. M., Theory and Implementation for the Sandia Waste-Isolution Flow and Transport Model for Fractured Media (SWIFT) Release 4.84 (1986)
[59] Richter, G. R., An optimal-order error estimate for the discontinuous Galerkin method, Math. Comp., 50, 75 (1988) · Zbl 0643.65059
[60] T. F. Russell, Eulerian-Lagrangian localized adjoint methods for advection-dominated problems, in, Pitmann Research Notes in Mathematics Series, edited by, Griffiths and Watson, Longman, Harlow, 1990, Vol, 228, p, 206.; T. F. Russell, Eulerian-Lagrangian localized adjoint methods for advection-dominated problems, in, Pitmann Research Notes in Mathematics Series, edited by, Griffiths and Watson, Longman, Harlow, 1990, Vol, 228, p, 206. · Zbl 0694.76037
[61] T. F. Russell, and, R. V. Trujillo, Eulerian-Lagrangian localized adjoint methods with variable coefficients in multiple dimensions, in, Computational Methods in Surface Hydrology, edited by, Gambolati, et al.; T. F. Russell, and, R. V. Trujillo, Eulerian-Lagrangian localized adjoint methods with variable coefficients in multiple dimensions, in, Computational Methods in Surface Hydrology, edited by, Gambolati, et al.
[62] T. F. Russell, and, M. F. Wheeler, Finite element and finite difference methods for continuous flows in porous media, in, The Mathematics of Reservoir Simulation, edited by, Ewing, Research Frontiers in Applied Mathematics, SIAM, Philadelphia, 1984, Vol, 1, p, 35.; T. F. Russell, and, M. F. Wheeler, Finite element and finite difference methods for continuous flows in porous media, in, The Mathematics of Reservoir Simulation, edited by, Ewing, Research Frontiers in Applied Mathematics, SIAM, Philadelphia, 1984, Vol, 1, p, 35.
[63] Schafer-Perini, A. L.; Wilson, J. L., Efficient and accurate front tracking for two-dimensional groundwater flow models, Water Resources Res., 27, 1471 (1991)
[64] Shu, C.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., 77, 439 (1988) · Zbl 0653.65072
[65] Smolarkiewicz, P. K., The multidimensional crowley advection scheme, Monthly Weather Rev., 110, 1968 (1982)
[66] Smoller, J., Shock Waves and Reaction-Diffusion Equations, 258 (1994) · Zbl 0807.35002
[67] E. Sunde, T. Thorstenson, T. Torsvik, J. E. Våg, and, M. S. Espedal, Field-related mathematical model to predict and reduce reservoir souring, in, Proceedings of SPE International Symposium on Oilfield Chemistry, New Orleans, 1993, SPE 25197.; E. Sunde, T. Thorstenson, T. Torsvik, J. E. Våg, and, M. S. Espedal, Field-related mathematical model to predict and reduce reservoir souring, in, Proceedings of SPE International Symposium on Oilfield Chemistry, New Orleans, 1993, SPE 25197.
[68] Sweby, P. K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21, 995 (1984) · Zbl 0565.65048
[69] Våg, J. E.; Wang, H.; Dahle, H. K., Eulerian-Lagrangian localized adjoint methods for systems of nonlinear advection-diffusion-reaction equations, Adv. Water Resources, 19, 297 (1996)
[70] van Leer, B., On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher, and Roe, SIAM J. Sci. Stat. Comp., 5, 1 (1984) · Zbl 0547.65065
[71] Varoglu, E.; Finn, W. D.L, Finite elements incorporating characteristics for one-dimensional diffusion-convection equation, J. Comput. Phys., 34, 371 (1980) · Zbl 0487.76083
[72] Wang, H., Eulerian-Lagrangian Localized Adjoint Methods: Analyses, Numerical Implementations and Applications (1992)
[73] Wang, H.; Al-Lawatia, M.; Telyakovskiy, S. A., A Runge-Kutta characteristic method for first-order linear hyperbolic equations, Numer. Methods PDEs, 13, 617 (1997) · Zbl 0904.65096
[74] Wang, H.; Ewing, R. E.; Celia, M. A., Eulerian-Lagrangian localized adjoint method for reactive transport with biodegradation, Numer. Methods PDEs, 11, 229 (1995) · Zbl 0824.76102
[75] H. Wang, R. C. Sharpley, and, S. Man, An ELLAM scheme for advection-diffusion equations in multi-dimensions, in, Computational Methods in Water Resources XI, edited by, Aldama, et al.; H. Wang, R. C. Sharpley, and, S. Man, An ELLAM scheme for advection-diffusion equations in multi-dimensions, in, Computational Methods in Water Resources XI, edited by, Aldama, et al.
[76] Westerink, J. J.; Shea, D., Consider higher degree Petrov-Galerkin methods for the solution of the transient convection-diffusion equation, Int. J. Numer. Methods Eng., 28, 1077 (1989) · Zbl 0679.76097
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.