The method of mixed volume element-characteristic mixed volume element and its numerical analysis for groundwater pollution in binary medium

Abstract

Groundwater pollution is an important topic of environmental sciences. Since the geologic structure is usually of crack-hole binary medium, its mathematical model is formulated by a nonlinear initial-boundary value problem of partial differential equations. The pressure is defined by an elliptic flow equation. The concentration of pollution is defined by a convection-diffusion equation. The surface adsorption concentration is defined by a first-order ordinary differential equation. The transport pressure appears within the concentration, and Darcy velocity controls the concentration. The flow equation is solved by the conservative mixed volume element and the computational accuracy of Darcy velocity is improved by one order. The mixed volume element with the characteristics is applied to approximate the concentration, i.e., the diffusion and convection are discretized by the method of mixed volume element and the characteristics, respectively. Sharp fronts are resolved stably by the characteristic discretizations without numerical dispersion or nonphysical oscillation. Large and accurate timesteps are used while the scheme has much smaller time truncation errors than those of standard methods on coarse grids. The mixed volume element is applied to approximate the diffusion. The concentration and its adjoint vector function are computed simultaneously, and the locally conservative law of mass is ensured. Using the theory and technique of a priori estimates of differential equations, we derive an optimal second order error in l2 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool in solving some actual applications.