An Upwind Multistep Difference with Mixed Finite Volume Element Method for a Positive Semi-Definite Contamination Treatment Problem (Analysis in Theory and Applications, 2025)
A positive semi-definite problem of three-dimensional incompressible contamination treatment from nuclear waste in porous media is discussed in this paper.The mathematical model is defined by a nonlinear initial-boundary system consistingof partial differential equations. Four important equations (an elliptic equation, twoconvection-diffusion equations and a heat conductor equation) determine the physicalfeatures. Considering the physical natures and computational efficiency, the authorsintroduce the conservative mixed finite volume element, upwind approximation andmultistep difference to solve this system. The pressure and Darcy velocity are computed by a mixed finite volume element. The concentrations and temperature aresolved by a combination of upwind approximation, multistep difference and mixedfinite volume element. A multistep difference is used for approximating the partialderivative with respect to time. Mixed finite volume element and upwind differencesare given for solving the convection-diffusions equations. Numerical dispersion andnonphysical oscillations could be eliminated, and the computational efficiency is improved by using a large time step. Furthermore, a conservative law is preserved anderror estimates in $L^2$-norm is obtained. Finally, two numerical experiments are givento show the efficiency and possible applications.
Keywords: Positive semi-definite problem; upwind multistep difference with mixed finite volume element; local conservation; convergence analysis; numerical experiments.