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Mixed Volume Element-Characteristic Fractional Step Difference Method for Contamination from Nuclear Waste Disposal

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Abstract

A nonlinear system with boundary-initial value conditions of convection–diffusion partial differential equations is presented to describe incompressible 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 pressure appears in convection–diffusion equations and heat equation in the form of Darcy velocity and controls the physical processes. The fluid pressure and velocity are solved by the conservative mixed volume element and the computation accuracy of Darcy velocity is improved one order. A combination method of the mixed volume element and the approximation of characteristics is applied to solve the brine and 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 strong computation stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. Larger time-steps along the characteristics are shown to result in smaller time-truncation errors than those resulting from standard methods. The mixed volume element method has the property of conservation on each element and it can obtain numerical solutions of the brine and adjoint vectors. The radionuclide is solved by a coupled method of characteristics and fractional step difference. The computational work is reduced greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and using the algorithm of speedup. Using numerical analysis of priori estimates of differential equations, we demonstrate an optimal second order estimate in \(l^2\) norm. Numerical data are appropriate with the scheme and it is shown that the method is a powerful tool to solve the well-known problems in porous media.

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Acknowledgements

The authors express their deep appreciation to Prof. J. Douglas Jr., Prof. R. E. Ewing, and Prof. Jiang Lishang for their many helpful suggestions in the serial research of numerical simulation of environmental sciences.

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Authors and Affiliations

  1. Institute of Mathematics, Shandong University, Jinan, 250100, China

    Changfeng Li, Yirang Yuan & Tongjun Sun

  2. School of Economics, Shandong University, Jinan, 250100, China

    Changfeng Li

  3. School of Mathematics, Shandong Normal University, Jinan, 250014, China

    Qing Yang

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  1. Changfeng Li
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  2. Yirang Yuan
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  3. Tongjun Sun
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  4. Qing Yang
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Correspondence to Yirang Yuan.

Additional information

Project supported by the National Natural Science Foundation of China (Grant Nos. 11101124 and 11271231), National Tackling Key Problems Program (Grant Nos. 2011ZX05052, 2011ZX05011-004).

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Li, C., Yuan, Y., Sun, T. et al. Mixed Volume Element-Characteristic Fractional Step Difference Method for Contamination from Nuclear Waste Disposal. J Sci Comput 72, 467–499 (2017). https://doi.org/10.1007/s10915-017-0365-3

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