We consider a nonlinear system with boundary-initial value conditions of convection-diffusion partial differential equations describing nuclear waste disposal contamination in porous media. The flow pressure is determined by an elliptic equation, the concentrations of brine and radionuclide are formulated by convection-diffusion equations, and the transport of temperature is defined by a heat equation. The transport pressure appears in the concentration equations and heat equation accompanying with Darcy velocity, and controls their processes. The flow equation is solved by the conservative method of mixed volume element and the accuracy of Darcy velocity is improved one order. The method of characteristic mixed volume element is applied to solve the concentrations and the heat, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm high computation stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while it has smaller time-truncation error and higher order of accuracy. The mixed volume element method has law of conservation on every element to treat the diffusion and it can obtain numerical solution is of the concentration and adjoint vectors. Using the theory and technique of priori estimate of differential equations, we derive an optimal second order estimate in l² norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve such an international famous problem.

nuclear waste disposal contamination in porous media, mixed volume element-characteristic mixed volume element, local conservation of mass, second order error in L² norm, numerical experiments.

Received: March 5, 2023; Accepted: July 6, 2023; Published: October 12, 2023

How to cite this article: Yirang Yuan, Changfeng Li, Yunxin Liu, Tongjun Sun and Qing Yang, Mixed volume element-characteristic mixed volume element of numerical simulation of nuclear contamination treatment and its convergence analysis, Far East Journal of Applied Mathematics 116(3) (2023), 263-310. http://dx.doi.org/10.17654/0972096023015

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] R. A. Adams, Sobolev Spaces, New York, Academic Press, 1975.
[2] T. Arbogast and M. F. Wheeler, A characteristics-mixed finite element methods for advection-dominated transport problems, SIAM J. Numer. Anal. 32(2) (1995), 404-424.
[3] J. B. Bell, C. N. Dawson and G. R. Shubin, An unsplit high-order Godunov scheme for scalar conservation laws in two dimensions, J. Comput. Phys. 74 (1988), 1-24.
[4] Z. Cai, On the finite volume element method, Numer. Math. 58 (1991), 713-735.
[5] Z. Cai, J. E. Jones, S. F. Mccormilk and T. F. Russell, Control-volume mixed finite element methods, Comput. Geosci. 1 (1997), 289-315.
[6] M. A. Cella, T. F. Russell, I. Herrera and R. E. Ewing, An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equations, Adv. Water Resour. 13(4) (1990), 187-206.
[7] S. H. Chou, D. Y. Kawk and P. Vassileviki, Mixed volume methods for elliptic problems on triangular grids, SIAM J. Numer. Anal. 35 (1998), 1850-1861.
[8] S. H. Chou, D. Y. Kawk and P. Vassileviki, Mixed volume methods on rectangular grids for elliptic problem, SIAM J. Numer. Anal. 37 (2000), 758-771.
[9] S. H. Chou and P. Vassileviki, A general mixed covolume frame work for constructing conservative schemes for elliptic problems, Math. Comp. 12 (2003), 150-161.
[10] 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(6) (1989), 1487-1512.
[11] J. Douglas Jr., Finite difference method for two-phase in compressible flow in porous media, SIAM J. Numer. Anal. 4 (1983), 681-696.
[12] J. Douglas Jr., R. E. Ewing and M. F. Wheeler, Approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numer. 17(1) (1983), 17-33.
[13] J. Douglas Jr., R. E. Ewing and M. F. Wheeler, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, RAIRO Anal. Numer. 17(3) (1983), 249-265.
[14] J. Douglas, Jr. and Y. R. Yuan, Numerical Simulation of Immiscible Flow in Porous Media based on Combining the Method of Characteristics with Mixed Finite Element Procedure, Numerical Simulation in Oil Recovery, Springer-Verlag, New York, 1986, pp. 119-132.
[15] R. E. Ewing, The Mathematics of Reservoir Simulation, SIAM, Philadelphia, 1983.
[16] R. E. Ewing, T. F. Russell and M. F. Wheeler, 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. Engrg. 47(1-2) (1984), 73-92.
[17] R. E. Ewing, Y. R. Yuan and G. Li, Finite element methods for contamination by nuclear waste-disposal in porous media, In Numerical Analysis, 1987, D. F. Griffiths and G. A. Watson, eds., Pitman Research Notes in Math. 1970, Longman Scientific and Technical, Fssex, U. K., 1988, pp. 53-66.
[18] R. E. Ewing, Y. R. Yuan and G. Li, A time-discretization procedure for a mixed finite element approximation of contamination by incompressible nuclear waste in porous media, Mathematics of Large Scale Computing, 127-146, New York and Basel: Marcel Dekker, INC, 1988.
[19] R. E. Ewing, Y. R. Yuan and G. Li, Time stepping along characteristics for a mixed finite element approximation for compressible flow of contamination from nuclear waste in porous media, SIAM J. Numer. Anal. 6 (1989), 1513-1524.
[20] L. S. Jiang and Z. Y. Pang, Finite Element Method and its Theory, Beijing: People’s Education Press, 1979.
[21] C. Johnson, Streamline Diffusion Methods for Problems in Fluid Mechanics, in Finite Element in Fluids VI, New York, Wiley, 1986.
[22] J. E. Jones, A Mixed Volume Method for Accurate Computation of Fluid Velocities in Porous Media, Ph.D. Thesis, University of Clorado, Denver, Co, 1995.
[23] R. H. Li and Z. Y. Chen, Generalized Difference of Differential Equations, Changchun, Jilin University Press, 1994.
[24] J. Nitsche, Linear splint-funktionen and die methoden von Ritz for elliptishce randwert probleme, Arch. for Rational Mech. and Anal. 36 (1968), 348-355.
[25] H. Pan and H. X. Rui, Mixed element method for two-dimensional Darcy-Forchheimer model, J. of Scientific Computing 52(3) (2012), 563-587.
[26] P. A. Raviart and J. M. Thomas, A Mixed Finite Element Method for Second Order Elliptic Problems, in: Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, 606, Springer, 1977.
[27] M. Reeves and R. M. Cranwall, User’s manual for the sanda waste-isolation flow and transport model (swift) release 4, 81, Sandia Report Nareg/CR-2324, SAND 81-2516, GF., 1981.
[28] H. X. Rui and H. Pan, A block-centered finite difference method for the Darcy-Forchheimer model, SIAM J. Numer. Anal. 50(5) (2012), 2612-2631.
[29] T. F. Russell, Time stepping along characteristics with incomplete interaction for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal. 22(5) (1985), 970-1013.
[30] T. F. Russell, Rigorous Block-centered Discretization on Inregular Grids: Improved simulation of complex reservoir systems, Project Report, Research Corporation, Tulsa, 1995.
[31] T. J. Sun and Y. R. Yuan, An Approximation of incompressible miscible displacement in porous media by mixed finite element method and characteristics-mixed finite element method, J. Comput. Appl. Math. 228(1) (2009), 391-411.
[32] M. R. Todd, P. M. O’Dell and G. J. Hirasaki, Methods for increased accuracy in numerical reservoir simulators, Soc. Petrol. Engry. J. 12(6) (1972), 521-530.
[33] A. Weiser and M. F. Wheeler, On convergence of block-centered finite difference for elliptic problems, SIAM J. Numer. Anal. 25(2) (1988), 351-375.
[34] D. P. Yang, Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems, Math. Comp. 69(231) (2000), 929 963.
[35] Y. R. Yuan, Numerical simulation and analysis for a model for compressible flow for nuclear waste-disposal contamination in porous media, Acta Mathematicae Applicatae Sinica 1 (1992), 70-82.
[36] Y. R. Yuan, Characteristic finite element methods for positive semidefinite problem of two phase miscible flow in three dimensions, Chin. Sci. Bull. 41(22) (1996), 2027-2032.
[37] Y. R. Yuan, Characteristic finite difference methods for positive semidefinite problem of two phase miscible flow in porous media, J. Systems Sci. Math. Sci. 12(4) (1999), 299-306.
[38] Y. R. Yuan, Theory and Application of Reservoir Numerical Simulation, Beijing, Science Press, 2013.