An Upwind Mixed Volume Element-Fractional Step Method on a Changing Mesh for Compressible Contamination Treatment from Nuclear Waste (Advances in Applied Mathematics and Mechanics, 2021)
In this paper the authors discuss the numerical simulation problem of three-dimensionalcompressible contamination treatment from nuclear waste. The mathematicalmodel is defined by an initial-boundary nonlinear convection-diffusion systemof four partial differential equations: a parabolic equation for the pressure, twoconvection-diffusion equations for the concentrations of brine and radionuclide and aheat conduction equation for the temperature. The pressure appears within the concentrationequations and heat conduction equation, and the Darcy velocity controls theconcentrations and the temperature. The pressure is solved by the conservative mixedvolume element method, and the order of the accuracy is improved by the Darcy velocity.The concentration of brine and temperature are computed by the upwind mixedvolume element method on a changing mesh, where the diffusion is discretized by amixed volume element and the convection is treated by an upwind scheme. The compositemethod can solve the convection-dominated diffusion problems well because iteliminates numerical dispersion and nonphysical oscillation and has high order computationalaccuracy. The mixed volume element has the local conservation of mass andenergy, and it can obtain the brine and temperature and their adjoint vector functionssimultaneously. The conservation nature plays an important role in numerical simulationof underground fluid. The concentrations of radionuclide factors are solved by themethod of upwind fractional step difference and the computational work is decreasedby decomposing a three-dimensional problem into three successive one-dimensionalproblems and using the method of speedup. By the theory and technique of a prioriestimates of differential equations, we derive an optimal order result in $L^2$ norm. Numericalexamples are given to show the effectiveness and practicability and the compositemethod is testified as a powerful tool to solve the well-known actual problem.