An upwind approximation combined with mixed volume element for a positive semi-definite contamination treatment from nuclear waste

Abstract

A positive semi-definite problem of three-dimensional contamination treatment from nuclear waste in porous media is discussed in this paper. The mathematical model is defined by a nonlinear initial-boundary system consisting of partial differential equations. An elliptic equation, two convection–diffusion equations and a heat conductor equation are given to determine the pressure, the concentrations of brine and radionuclide, and the temperature, respectively. The parabolic equations include Darcy velocity dependent on the pressure, and the pressure affects the whole physical process. A conservative mixed volume element is used. One-order computational accuracy is improved for Darcy velocity. The main physical unknowns are solved by an upwind-mixed volume element, where the diffusions and convection terms are treated by a mixed volume element and an upwind approximation, respectively. This composite method could eliminate numerical dispersion and nonphysical oscillations, thus this problem may be solved well. The mixed volume element can obtain the concentrations, temperature and their adjoint vectors simultaneously, and has the conservation of mass or energy, an important law in numerical simulation. Applying the theory and special treatment of a priori estimates, we obtain optimal order estimates in $$L^2$$ -norm. Numerical experiments show the efficiency and applicability for solving such complicated problems.