A Comparative Study of Finite Element and Finite Difference Methods for Two-Dimensional Space-Fractional Advection-Dispersion Equation

2015 ◽  
Vol 8 (1) ◽  
pp. 166-186 ◽  
Author(s):  
Guofei Pang ◽  
Wen Chen ◽  
Kam Yim Sze
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Comparative Study ◽  
Dispersion Equation ◽  
Dimensional Space ◽  
Finite Difference Methods ◽  
Rectangular Domain ◽  
Two Dimensional ◽  
Element Analysis ◽  
Advection Dispersion

AbstractThe paper makes a comparative study of the finite element method (FEM) and the finite difference method (FDM) for two-dimensional fractional advection-dispersion equation (FADE) which has recently been considered a promising tool in modeling non-Fickian solute transport in groundwater. Due to the non-local property of integro-differential operator of the space-fractional derivative, numerical solution of FADE is very challenging and little has been reported in literature, especially for high-dimensional case. In order to effectively apply the FEM and the FDM to the FADE on a rectangular domain, a backward-distance algorithm is presented to extend the triangular elements to generic polygon elements in the finite element analysis, and a variable-step vector Grünwald formula is proposed to improve the solution accuracy of the conventional finite difference scheme. Numerical investigation shows that the FEM compares favorably with the FDM in terms of accuracy and convergence rate whereas the latter enjoys less computational effort.

Numerical treatment for solving two-dimensional space-fractional advection–dispersion equation using meshless method

2018 ◽  
Vol 32 (06) ◽  
pp. 1850073 ◽  
Author(s):  
Rongjun Cheng ◽  
Fengxin Sun ◽  
Qi Wei ◽  
Jufeng Wang
Keyword(s):  
Dispersion Equation ◽  
Convergence Rates ◽  
Dimensional Space ◽  
Energy Functional ◽  
Least Square ◽  
Approximation Schemes ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Moving Least Square ◽  
Advection Dispersion

Space-fractional advection–dispersion equation (SFADE) can describe particle transport in a variety of fields more accurately than the classical models of integer-order derivative. Because of nonlocal property of integro-differential operator of space-fractional derivative, it is very challenging to deal with fractional model, and few have been reported in the literature. In this paper, a numerical analysis of the two-dimensional SFADE is carried out by the element-free Galerkin (EFG) method. The trial functions for the SFADE are constructed by the moving least-square (MLS) approximation. By the Galerkin weak form, the energy functional is formulated. Employing the energy functional minimization procedure, the final algebraic equations system is obtained. The Riemann–Liouville operator is discretized by the Grünwald formula. With center difference method, EFG method and Grünwald formula, the fully discrete approximation schemes for SFADE are established. Comparing with exact results and available results by other well-known methods, the computed approximate solutions are presented in the format of tables and graphs. The presented results demonstrate the validity, efficiency and accuracy of the proposed techniques. Furthermore, the error is computed and the proposed method has reasonable convergence rates in spatial and temporal discretizations.

Finite difference methods for two-dimensional fractional dispersion equation

2006 ◽  
Vol 211 (1) ◽  
pp. 249-261 ◽  
Author(s):  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler ◽  
Charles Tadjeran
Keyword(s):  
Finite Difference ◽  
Dispersion Equation ◽  
Finite Difference Methods ◽  
Two Dimensional ◽  
Difference Methods

scholarly journals Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deformations

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1523 ◽  
Author(s):  
Daria Scerrato ◽  
Ivan Giorgio
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Variational Formulation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Element Analysis ◽  
Elastic Continuum ◽  
Bias Extension ◽  
Three Dimensional Space

A particular pantographic sheet, modeled as a two-dimensional elastic continuum consisting of an orthogonal lattice of continuously distributed fibers with a cycloidal texture, is introduced and investigated. These fibers conceived as embedded beams on the surface are allowed to be deformed in a three-dimensional space and are endowed with resistance to stretching, shearing, bending, and twisting. A finite element analysis directly derived from a variational formulation was performed for some explanatory tests to illustrate the behavior of the newly introduced material. Specifically, we considered tests on: (1) bias extension; (2) compressive; (3) shear; and (4) torsion. The numerical results are discussed to some extent. Finally, attention is drawn to a comparison with other kinds of orthogonal lattices, namely straight, parabolic, and oscillatory, to show the differences in the behavior of the samples due to the diverse arrangements of the fibers.

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