Multistep characteristic method for incompressible flow in porous media. (English) Zbl 1166.76038

Long, Xiaohan; Yuan, Yirang

Appl. Math. Comput. 214, No. 1, 259-270 (2009).

Summary: We present a multistep difference scheme for the problem of miscible displacement of incompressible fluid flow in porous media. The discretization involves a three-level time scheme based on the characteristic method, and a five-point finite difference scheme for space discretization. We prove that the convergence is of order \(O(h^{2}+(\Delta t)^{2})\), which is in contrast to the convergence of order \(O(h+\Delta t)\) proved for a singlestep characteristic with the same space discretization. Numerical experiments demonstrate the stability and second-order convergence of the scheme.

MSC:

76M20	Finite difference methods applied to problems in fluid mechanics
76S05	Flows in porous media; filtration; seepage
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Keywords:

five-point finite difference scheme; second-order convergence

References:

[1]	Bermúdez, A.; Nogueiras, M. R.; Vázquez, C., Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements. Part I: Time discretization, SIAM J. Numer. Anal., 44, 1829-1853 (2006) · Zbl 1126.65080
[2]	Bermúdez, A.; Nogueiras, M. R.; Vázquez, C., Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements. Part II: Fully discretized scheme and quadrature formulas, SIAM J. Numer. Anal., 44, 1854-1876 (2006) · Zbl 1126.65081
[3]	Bermejo, R., On the equivalence of semi-Lagrange schemes and particle-in-cell finite element methods, Mon. Weather Rev., 118, 979-987 (1990)
[4]	Boukir, K.; Maday, Y.; Métivet, B., A high order characteristics method for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 116, 211-218 (1994) · Zbl 0824.76060
[5]	Boukir, K.; Maday, Y.; Métivet, B.; Razafindrakoto, E., A high-order characteristics/finite element method for the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 25, 1421-1454 (1997) · Zbl 0904.76040
[6]	Bramble, J. H., A second order finite difference analog of the first biharmonic boundary value problem, Numer. Math., 4, 236-249 (1966) · Zbl 0154.41105
[7]	Celia, M. A.; Russell, T. F.; Herrera, I.; Ewing, R. E., An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation, Adv. Water Res., 13, 187-206 (1990)
[8]	Douglas, J.; Ewing, R. E.; Wheeler, M. F., The approximation of the pressure by a mixed method in the simulation of immiscible displacement, RAIRO Anal. Numer., 17, 17-33 (1983) · Zbl 0516.76094
[9]	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, 871-885 (1982) · Zbl 0492.65051
[10]	Douglas, J., Finite difference methods for two-phase incompressible flow in porous media, SIAM J. Numer. Anal., 20, 681-696 (1983) · Zbl 0519.76107
[12]	Ewing, R. E.; Russell, T. F.; Wheeler, M. F., Convergence analysis of an approximation of immiscible displacement in porous media by mixed finite elements and a modified method of characteristics, Methods Appl. Mech. Eng., 47, 73-92 (1984) · Zbl 0545.76131
[13]	Ewing, R. E.; Russell, T. F.; Wheeler, M. F., Simulation of immiscible displacement using mixed methods and a modified method of characteristics, SPE, 12241, 71-81 (1983)
[15]	Fourestey, G.; Piperno, S., A second-order time-accurate ALE Lagrange-Galerkin method applied to wind engineering and control of bridge profiles, Comput. Methods Appl. Mech. Eng., 193, 4117-4137 (2004) · Zbl 1175.76101
[16]	Hairer, E.; Wanner, H., Solving Ordinary Differential Equations I - Nonstiff Problems (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0638.65058
[17]	Süli, E., Convergence and nonlinear stability of the Lagrange-Galerkin method for Navier-Stokes Equations, Numer. Math., 53, 459-483 (1988) · Zbl 0637.76024
[18]	Pironneau, O., On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math., 38, 309-332 (1982) · Zbl 0505.76100
[19]	Rui, H.; Tabata, M., A second order characteristic finite element scheme for convection-diffusion problems, Numer. Math., 92, 161-177 (2002) · Zbl 1005.65114
[20]	Russell, T. F., Time stepping along characteristics with incomplete iteration for a Galerkin approximation of immiscible displacement in porous media, SIAM J. Numer. Anal., 22, 970-1013 (1985) · Zbl 0594.76087
[21]	Wang, H.; Dahle, H. K.; Ewing, R. E.; Espedal, M. S.; Sharpley, R. C.; Man, S., An ELLAM scheme for advection-diffusion equations in two dimensions, SIAM J. Sci. Comput., 20, 2160-2194 (1999) · Zbl 0939.65109
[22]	Wang, H.; Liang, D.; Ewing, R. E.; Lyons, S. L.; Qin, G., An improved numerical simulator for different types of flows in porous media, Numer. Methods Part. Diff. Eq., 19, 343-362 (2003) · Zbl 1079.76044
[23]	Xiu, X.; Karniadakis, G. E., A semi-Lagrangian high-order method for Navier-Stokes equations, J. Comput. Phys., 172, 658-684 (2001) · Zbl 1028.76026
[24]	Yirang, Yuan, The characteristic-mixed finite element method for enhanced oil recovery simulation and optimal order \(L^2\) error estimate, Chinese Sci. Bull., 38, 1066-1070 (1993) · Zbl 0806.76048
[25]	Zlámal, M., Finite element methods for nonlinear parabolic equations, RAIRO Numer. Anal., 11, 93-107 (1977) · Zbl 0385.65049