Finite difference approximations for fractional advection-dispersion flow equations

By Mark M. Meerschaert, Charles Tadjeran. 2004

1,519 citations recorded by the source corpus.

Subfields: Mechanics of Materials, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/a35cae9a0fb6f6ec61d39125710c0b88213c7048

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In Mechanics of Materials

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

An implicit RBF meshless approach for time fractional diffusion equations (2011, methodology): “At present, most of FPDEs are solved numerically by Finite Difference Method (FDM) [10,11], when a few of research has been reported using Finite Element Method (FEM) [12–14].”
THE APPROXIMATE AND EXACT SOLUTIONS OF THE FRACTIONAL-ORDER DELAY DIFFERENTIAL EQUATIONS USING LEGENDRE SEUDOSPECTRAL METHOD (2012, background): “Most fractional differential equations do not have exact solutions, so approximation and numerical techniques (see [3-7], [13], [21]), must be used.”
Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients (2015, methodology): “In the last decade or so, extensive research has been carried out on the development of numerical methods for fractional partial differential equations, including finite difference method [11–13], finite element methods [14, 15], and spectral methods [16].”
A class of second order difference approximations for solving space fractional diffusion equations (2012, methodology): “The Riemann-Liouville fractional derivative can be discretized by the standard Grünwald-Letnikov form ula [18] with only the first order accuracy, but the difference scheme based on the Grünwald-Letnikov form ula for time dependent problems is unstable [12].”
Implicit difference approximation for the time fractional diffusion equation (2006, background): “Meerschaert et al. proposed finite difference approximations for two-sided space-fractional partial dierential equations [15] and fractional advection-dispersion flow equations [ 16 ].”
A finite element solution for the fractional advection–dispersion equation (2008, methodology): “Besides, some numerical solutions of the FADE only take the left-handed fractional derivative into account [11,23,25,49,50], implying that the backward transition of solute particles that is slower than the average velocity is eliminated.”
A second-order accurate numerical approximation for the fractional diffusion equation (2006, methodology): “The standard Grünwald estimates generally yield unstable finite difference equations regardless of whether the resulting finite difference method is an explicit or an implicit system, see [9] for related discussion.”
Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems (2013, methodology): “(4), also (1), can be obtained by the use of Grunwald-Letnikov approximation of the fractional order derivative [3].”
A characteristic difference method for the variable-order fractional advection-diffusion equation (2013, methodology): “Using the relationship between Riemann-Liouville and Grünwald-Letnikov derivatives, two discrete approximations to the space fractional derivative terms may be defined from the shifted Grünwald-formula [19, 20].”
Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation (2007, methodology): “For the space fractional derivatives Dx uðxi; tkþ1Þ and Dxuðxi; tkþ1Þ, we adopted the Grünwald formula [11] and the shift Grünwald formula at level tk+1 [7], respectively Dx uðxi; tkþ1Þ 1⁄4 1 h Xi j1⁄40 g j uðxi jh; tkþ1Þ þOðhÞ; ð9Þ Dxuðxi; tkþ1Þ 1⁄4 1 h Xiþ1 j1⁄40 g j uðxi ðj 1Þh; tkþ1Þ þOðhÞ; ð10Þ”
A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations (2015, methodology): “The space fractional advection–dispersion equation (SFADE) is used to describe the transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns [14,15] and it is also proved as a benefit tool to model the transport of passive tracers carried”
Compact finite difference method for the fractional diffusion equation (2009, methodology): “Meerschaert and Tadjeran [4] studied the onedimensional radial flow model, they found that the fractional derivative describes more accurately the early arrival that cannot be explained by the classical advection–dispersion equations and they presented an implicit Euler method, based on a .”
A second-order accurate numerical method for the two-dimensional fractional diffusion equation (2007, background): “[12,13,15,21] treat explicit or implicit finite difference methods for space fractional diffusion equations, and [9,29] examine finite element methods for such problems.”