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Abstract
We develop a finite volume characteristic method for the solution of the advection-diffusion equations which model the contaminant transport through porous medium. This method uses a second order Runge-Kutta approximation for the characteristics within the framework of the Eulerian Lagrangian localized adjoint methods (ELLAM). The derived scheme conserves mass, symmetrizes the governing equations and generates accurate numerical solutions even if large time steps are used. Numerical experiments comparing several competitive methods using a standard test example are presented to illustrate the performance of the method.
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References
- Al-Lawatia, M. Sharpley, R.C. and Wang, H. 1999. Second-order characteristics methods for advection-diffusion equations and comparison to other schemes, Advances in Water Resources, 22(7):741-768.
- Boris, J.P. and Book, D.L., 1997. Flux-Corrected Transport. I. SHASTA, A Fluid Transport Algorithm That Works, Journal of Computational Physics,135(2):172-186.
- Bouloutas E.T. and Celia, M.A. 1991. An improved cubicPetrov-Galerkin method for simulation of transient advection-diffusion processes in rectangularly decomposable domains, Comp. Meth. Appl. Mech. Engrg. 91: 289-308.
- Celia, M.A. Russell, T.F. Herrera, I. and Ewing, R.E. 1990. An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation, Advances in Water Resources 13:187-206.
- Dahle, H.K. Ewing, R.E. and Russell, T.F. 1995. Eulerian-Lagrangian localized adjoint methods for a nonlinear advection-diffusion equation, Comp. Meth. Appl. Mech. Engrg. 122:223-250.
- Dawson, C.N. 1991. Godunov-mixed methods for advective flow problems in one space dimension, SIAM J. Numer. Anal. 28:1282-1309.
- Douglas, J. Jr. and 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. 19:871-885, 1982.
- Healy R.W. and Russel, T.F. 1993. A finite-volume Eulerian-Lagrangian localized adjoint method for solution of the advection-dispersion equation, Water Resour. Res. 29:2399-2413.
- Hughes T.J.R. and Mallet, M. 1986. A new finite element formulation for computational fluid dynamics III, The general Streamline operator for multidimensional advective-diffusive systems, Comp. Meth. Appl. Mech. Engrg. 58:305-328.
- Johnson, C. 1987. Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge.
- Pironneau, O. 1982. On the transport-diffusion algorithmand its application to the Navier-Stokes equations, Numer. Math. 38:309-332.
- Russell T.F. and Trujillo, R.V. 1990. Eulerian-Lagrangian localized adjoint methods with variable coefficients in multiple dimensions, in: Gambolati, et al., eds., Computational Methods in Surface Hydrology 357-363, Springer-Verlag, Berlin.
- Shu, C-W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in: A. Quarteroni, ed., Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, CIME subseries, Springer-Verlag, to appear.
- Tóth, G. and Odstrčil, D. 1996. Comparison of Some Flux Corrected Transport and Total Variation Diminishing Numerical Schemes for Hydrodynamic and Magnetohydrodynamic Problems, Journal of Computational Physics, 128(1): 82-100.
- Van Leer, B. 1984. On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe, SIAM J. Sci. Statist. Comput. 5:1-20.
- Wang, H. Ewing, R.E. and Russell, T.F., 1992. ELLAM for variable-coefficient convection-diffusion problems arising in groundwater applications, in: Russell et al. (eds.), Computational Methods in Water Resources IX, Vol. I, Computational Mechanics Publications and Elsevier Applied Science, London and New York, 25-31.
- Westerink, J.J. and Shea, D. 1989. Consistent higher degree Petrov-Galerkin methods for the solution of the transient convection-diffusion equation, Int. J. Numer. Meth. Engrg. 28:1077-1101.
References
Al-Lawatia, M. Sharpley, R.C. and Wang, H. 1999. Second-order characteristics methods for advection-diffusion equations and comparison to other schemes, Advances in Water Resources, 22(7):741-768.
Boris, J.P. and Book, D.L., 1997. Flux-Corrected Transport. I. SHASTA, A Fluid Transport Algorithm That Works, Journal of Computational Physics,135(2):172-186.
Bouloutas E.T. and Celia, M.A. 1991. An improved cubicPetrov-Galerkin method for simulation of transient advection-diffusion processes in rectangularly decomposable domains, Comp. Meth. Appl. Mech. Engrg. 91: 289-308.
Celia, M.A. Russell, T.F. Herrera, I. and Ewing, R.E. 1990. An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation, Advances in Water Resources 13:187-206.
Dahle, H.K. Ewing, R.E. and Russell, T.F. 1995. Eulerian-Lagrangian localized adjoint methods for a nonlinear advection-diffusion equation, Comp. Meth. Appl. Mech. Engrg. 122:223-250.
Dawson, C.N. 1991. Godunov-mixed methods for advective flow problems in one space dimension, SIAM J. Numer. Anal. 28:1282-1309.
Douglas, J. Jr. and 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. 19:871-885, 1982.
Healy R.W. and Russel, T.F. 1993. A finite-volume Eulerian-Lagrangian localized adjoint method for solution of the advection-dispersion equation, Water Resour. Res. 29:2399-2413.
Hughes T.J.R. and Mallet, M. 1986. A new finite element formulation for computational fluid dynamics III, The general Streamline operator for multidimensional advective-diffusive systems, Comp. Meth. Appl. Mech. Engrg. 58:305-328.
Johnson, C. 1987. Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge.
Pironneau, O. 1982. On the transport-diffusion algorithmand its application to the Navier-Stokes equations, Numer. Math. 38:309-332.
Russell T.F. and Trujillo, R.V. 1990. Eulerian-Lagrangian localized adjoint methods with variable coefficients in multiple dimensions, in: Gambolati, et al., eds., Computational Methods in Surface Hydrology 357-363, Springer-Verlag, Berlin.
Shu, C-W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in: A. Quarteroni, ed., Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, CIME subseries, Springer-Verlag, to appear.
Tóth, G. and Odstrčil, D. 1996. Comparison of Some Flux Corrected Transport and Total Variation Diminishing Numerical Schemes for Hydrodynamic and Magnetohydrodynamic Problems, Journal of Computational Physics, 128(1): 82-100.
Van Leer, B. 1984. On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe, SIAM J. Sci. Statist. Comput. 5:1-20.
Wang, H. Ewing, R.E. and Russell, T.F., 1992. ELLAM for variable-coefficient convection-diffusion problems arising in groundwater applications, in: Russell et al. (eds.), Computational Methods in Water Resources IX, Vol. I, Computational Mechanics Publications and Elsevier Applied Science, London and New York, 25-31.
Westerink, J.J. and Shea, D. 1989. Consistent higher degree Petrov-Galerkin methods for the solution of the transient convection-diffusion equation, Int. J. Numer. Meth. Engrg. 28:1077-1101.