A reliable algorithm to determine the pollution transport within underground reservoirs: implementation of an efficient collocation meshless method based on the moving Kriging interpolation

The pollution propagation within the underground water reservoirs is a challenging and important phenomenon. In the current work, the numerical simulation of pollution transport in an underground channel is performed using the meshless method. To account the anomalous dispersion in a general case, the variable order fractional mass transfer equation is utilized for a rectangular channel. The clean fluid stream enters the channel and due to several phenomenon including the leakage of pollution from the channel walls, the internal pollution source, and the occurrence of the chemical reactions, the pollution content is affected. The non-dimensional form of the governing equation is derived to introduce the dominant dimensionless group numbers. The numerical solution of the obtained equation is established based on the meshless local Petrov–Galerkin method using the moving Kriging interpolation. The Dirac delta function is used as a test function over the local sub-domains. To discretize the present formulation in space variables, we apply the moving Kriging shape functions. Also, to estimate the fractional-order versus the time, finite difference relation is utilized. Using Kronecker’s delta property of moving Kriging interpolation shape functions the boundary conditions in the final system are imposed automatically. The main aim of this technique is to investigate a global estimation for the model, which consequently decrease such problems to those of solving a system of algebraic equations. To determine the accuracy and efficiency of the present method on regular and irregular domains, an example is given in various domains and with regular and irregular distributed points. Also, the effect of major parameters including the fractional order exponent, leakage velocity, chemical reaction rate constant, diffusion coefficient in addition to the stationary/moving pollution source is also examined. It will be shown that, by enhancement of the diffusivity from 0.1 to 20, the outlet concentration reduces by 25.1%, while diffusivity increase from 20 to 50 affects the exiting pollution by merely 7.0%.

[1]  Zakieh Avazzadeh,et al.  A computational wavelet method for variable-order fractional model of dual phase lag bioheat equation , 2019, J. Comput. Phys..

[2]  Guirong Liu,et al.  A point interpolation method for two-dimensional solids , 2001 .

[3]  Adem Kilicman,et al.  Application of Homotopy Perturbation and Variational Iteration Methods for Fredholm Integrodifferential Equation of Fractional Order , 2012 .

[4]  A. Shokri,et al.  A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation , 2016 .

[5]  M. H. Akbari,et al.  Lean flammability limits for stable performance with a porous burner , 2009 .

[6]  Jiye Yang,et al.  Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations , 2015, J. Comput. Phys..

[7]  J. Machado,et al.  A new fractional operator of variable order: Application in the description of anomalous diffusion , 2016, 1611.09200.

[8]  Carlo Cattani,et al.  Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations , 2014, Commun. Nonlinear Sci. Numer. Simul..

[9]  Akira Hasegawa,et al.  Optical solitons in fibers , 1993, International Commission for Optics.

[10]  Y. Duan,et al.  Meshless Radial Basis Function Method for Transient Electromagnetic Computations , 2008, IEEE Transactions on Magnetics.

[11]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[12]  M. Heydari,et al.  A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation , 2019, Computer Methods in Applied Mechanics and Engineering.

[13]  Xiaoting Liu,et al.  Simulating multi-dimensional anomalous diffusion in nonstationary media using variable-order vector fractional-derivative models with Kansa solver , 2019 .

[14]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[15]  Kang Tai,et al.  Radial point interpolation collocation method (RPICM) for the solution of nonlinear poisson problems , 2005 .

[16]  Shiyou Yang,et al.  Application of a meshless method in electromagnetics , 2001 .

[17]  K. M. Liew,et al.  A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems , 2011 .

[18]  M. D. dos Santos.  Analytic approaches of the anomalous diffusion: A review , 2019, Chaos, Solitons & Fractals.

[19]  M. Heydari,et al.  Numerical study of unsteady natural convection of variable-order fractional Jeffrey nanofluid over an oscillating plate in a porous medium involved with magnetic, chemical and heat absorption effects using Chebyshev cardinal functions , 2019, The European Physical Journal Plus.

[20]  Mohammad Reza Hooshmandasl,et al.  An efficient computational method for solving fractional biharmonic equation , 2014, Comput. Math. Appl..

[21]  Guirong Liu,et al.  A LOCAL RADIAL POINT INTERPOLATION METHOD (LRPIM) FOR FREE VIBRATION ANALYSES OF 2-D SOLIDS , 2001 .

[22]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[23]  The transport dynamics in complex systems governing by anomalous diffusion modelled with Riesz fractional partial differential equations , 2017 .

[24]  S. Abbasbandy,et al.  MLPG method for two-dimensional diffusion equation with Neumann's and non-classical boundary conditions , 2011 .

[25]  L. Gu,et al.  Moving kriging interpolation and element‐free Galerkin method , 2003 .

[26]  Baodong Dai,et al.  A meshless local moving Kriging method for two-dimensional solids , 2011, Appl. Math. Comput..

[27]  S. Atluri,et al.  The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods , 2002 .

[28]  Minimizing sequences for differential inclusion problems in the presence of state constraints , 1990 .

[29]  Yonghong Wu,et al.  Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion , 2017, Appl. Math. Lett..

[30]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[31]  M. Heydari,et al.  An efficient meshless method based on the moving Kriging interpolation for two‐dimensional variable‐order time fractional mobile/immobile advection‐diffusion model , 2020, Mathematical Methods in the Applied Sciences.

[32]  Zhi‐zhong Sun,et al.  A compact difference scheme for the fractional diffusion-wave equation , 2010 .

[33]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[34]  Michael Griebel,et al.  Meshfree Methods for Partial Differential Equations IV , 2005 .

[35]  Jian-feng Zhang.  Environmental Problems of Human Settlements and Countermeasures Based on Ecological Engineering , 2019 .

[36]  Carlo Cattani,et al.  Wavelets method for solving fractional optimal control problems , 2016, Appl. Math. Comput..

[37]  Guirong Liu,et al.  A point interpolation meshless method based on radial basis functions , 2002 .

[38]  S. Abbasbandy,et al.  Numerical solution of a modified anomalous diffusion equation with nonlinear source term through meshless singular boundary method , 2019, Engineering Analysis with Boundary Elements.