Mixed volume element-characteristic mixed volume element of numerical simulation of nuclear contamination treatment and its convergence analysis

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.