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Article Dans Une Revue Communications in Nonlinear Science and Numerical Simulation Année : 2021

An investigation of nonlinear time-fractional anomalous diffusion models for simulating transport processes in heterogeneous binary media

Résumé

In this work, we consider two of the most frequently used two-dimensional nonlinear time-fractional anomalous sub-diffusion models for simulating transport phenomena in heterogeneous binary media—a variable-order model and a generalised transport equation based on the Riemann-Liouville fractional operator. Our computational modelling frame- work uses a second order modified weighted shifted Grünwald-Letnikov scheme with cor- rection terms to approximate the time-fractional derivative and nonlinear source term, to- gether with an unstructured mesh control volume method for the spatial discretisation that accommodates the heterogeneous model properties. The resulting nonlinear system of algebraic equations is then stepped in time using an efficient Jacobian-free, Newton-Krylov solver to determine the set of discrete solution unknowns. We derive a semi-analytical solution and mass balance equation for a class of two-layered problems to evaluate the accuracy of the different computational models. A key contribution of our work is to iden- tify the correct form of the interfacial boundary conditions to impose for the flux terms within the time-fractional framework and to illustrate the significant impact that the time- fractional indices have on the mass transfer. We show that the generalised transport model not only exhibits the correct physical solution behaviour, it produces a more accurate over- all mass balance in comparison to the variable-order time-fractional model which is un- able to resolve the abrupt changes in the solution behaviour at the interface between two different media having contrasting diffusivity, fractional index and source term. This find- ing indicates that this generalised model is an effective tool for characterising anomalous transport processes in heterogeneous systems. Finally, a series of numerical examples are presented to verify the theoretical analysis and to highlight the capability of the gener- alised transport model.
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Dates et versions

hal-03141518 , version 1 (15-02-2021)

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Libo Feng, Ian Turner, Patrick Perré, Kevin Burrage. An investigation of nonlinear time-fractional anomalous diffusion models for simulating transport processes in heterogeneous binary media. Communications in Nonlinear Science and Numerical Simulation, 2021, 92, pp.105454. ⟨10.1016/j.cnsns.2020.105454⟩. ⟨hal-03141518⟩
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