Time-dependent fractional advection-diffusion equations by an implicit MLS meshless method. (English) Zbl 1242.76262
Summary: Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in an FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is finite difference method (FDM), which is usually difficult to handle a complex problem domain, and also difficult to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection-diffusion equations (FADE), which is a typical FPDE. The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong-forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate the accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE.
MSC:
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76R99 | Diffusion and convection |
Keywords:
fractional partial differential equation; fractional advection; diffusion equation; moving least squares; meshless method; meshfree method; implicit numerical schemesSoftware:
Mfree2DReferences:
| [1] | Agrawal, Introduction (Preface), Nonlinear Dynamics 38 pp 1– (2004) · doi:10.1007/s11071-004-3743-y |
| [2] | Benson, Application of a fractional advection-dispertion equation, Water Resources Research 36 (6) pp 1403– (2000) · doi:10.1029/2000WR900031 |
| [3] | Butzer PL Westphal U An Introduction to Fractional Calculus World Scientific Singapore 1 85 |
| [4] | Miller, An Introduction to the Fractional Calculus and Fractional Differential Equations (1993) · Zbl 0789.26002 |
| [5] | Metzler, Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck approach, Physics Review Letter 82 pp 3563– (1999) · doi:10.1103/PhysRevLett.82.3563 |
| [6] | Podlubny, Fractional Differential Equations (1999) |
| [7] | Chechkin, Fundamentals of Lévy fight processes, Advances in Chemical Physics 133 pp 436– (2006) |
| [8] | Metzler, The random walk’s guide to anomalous diffusion a fractional dynamics approach, Physics Reports 339 (1) pp 1– (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3 |
| [9] | Sokolov, Fractional kinetic equations, Physics Today 55 pp 48– (2002) · doi:10.1063/1.1535007 |
| [10] | Zaslavsky, Chao, fractional kinetics, and anomalous transport, Physics Reports 371 (1) pp 461– (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9 |
| [11] | Metzler, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of Physics A: Mathematical and General 37 (2) pp R161– (2004) · Zbl 1075.82018 · doi:10.1088/0305-4470/37/31/R01 |
| [12] | Meerschaert, Finite difference approximations for fractional advection-dispersion flow equations, Journal of Computational and Applied Mathematics 172 pp 65– (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033 |
| [13] | Liu, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Applied Mathematics and Computation 191 pp 12– (2007) · Zbl 1193.76093 · doi:10.1016/j.amc.2006.08.162 |
| [14] | Zhuang, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM Journal on Numerical Analysis 47 (3) pp 1760– (2009) · Zbl 1204.26013 · doi:10.1137/080730597 |
| [15] | Liu, Numerical simulation for the 3D seepage flow with fractional derivatives in porous media, IMA Journal of Applied Mathematics 74 (2) pp 201– (2008) · Zbl 1169.76427 · doi:10.1093/imamat/hxn044 |
| [16] | Yang Q Novel analytical and numerical methods for solving fractional dynamical systems 2010 |
| [17] | Ervin, Variational formulation for the stationary fractional advection-dispersion equation, Numerical Methods Partial Differential Equations 22 pp 558– (2004) · Zbl 1095.65118 · doi:10.1002/num.20112 |
| [18] | Ervin, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM Journal on Numerical Analysis 45 pp 572– (2007) · Zbl 1141.65089 · doi:10.1137/050642757 |
| [19] | Roop, Computational aspect of FEM approximation of fractional advection dispersion equation on bounded domains in R2, Journal of Computational and Applied Mathematics 193 (1) pp 243– (2006) · Zbl 1092.65122 · doi:10.1016/j.cam.2005.06.005 |
| [20] | Lin, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics 225 (2) pp 1533– (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 |
| [21] | Belytschko, Meshless methods: overview and recent developments, Computer Methods in Applied Mechanics and Engineering 139 pp 3– (1996) · Zbl 0891.73075 · doi:10.1016/S0045-7825(96)01078-X |
| [22] | Onate, A finite method in computational mechanics: applications to convective transport and fluid flow, International Journal for Numerical Methods in Engineering 39 pp 3839– (1996) · Zbl 0884.76068 · doi:10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R |
| [23] | Gingold, Smooth particle hydrodynamics: theory and applications to non spherical stars, Monthly Notices of the Royal Astronomical Society 181 pp 375– (1977) · Zbl 0421.76032 · doi:10.1093/mnras/181.3.375 |
| [24] | Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics, Computers and Mathematics with Applications 19 pp 127– (1990) · Zbl 0692.76003 · doi:10.1016/0898-1221(90)90270-T |
| [25] | Nayroles, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics 10 (5) pp 307– (1992) · Zbl 0764.65068 · doi:10.1007/BF00364252 |
| [26] | Liu, Reproducing kernel particle methods, International Journal for Numerical Methods in Engineering 20 pp 1081– (1995) · Zbl 0881.76072 · doi:10.1002/fld.1650200824 |
| [27] | Liu, A point interpolation method for two-dimensional solids, International Journal for Numerical Methods in Engineering 50 (4) pp 937– (2001) · Zbl 1050.74057 · doi:10.1002/1097-0207(20010210)50:4<937::AID-NME62>3.0.CO;2-X |
| [28] | Liu, Assessment and applications of point interpolation methods for computational mechanics, International Journal for Numerical Methods in Engineering 59 (10) pp 1373– (2004) · Zbl 1041.74562 · doi:10.1002/nme.925 |
| [29] | Atluri, A new meshfree local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics 22 pp 117– (1998) · Zbl 0932.76067 · doi:10.1007/s004660050346 |
| [30] | Liu, A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids, Journal of Sound and Vibration 246 (1) pp 29– (2001) · doi:10.1006/jsvi.2000.3626 |
| [31] | Liu, Comparisons of two meshfree local point interpolation methods for structural analyses, Computational Mechanics 29 (2) pp 107– (2002) · Zbl 1053.74659 · doi:10.1007/s00466-002-0320-4 |
| [32] | Mukherjee, Boundary node method for potential problems, International Journal for Numerical Methods in Engineering 40 pp 797– (1997) · Zbl 0885.65124 · doi:10.1002/(SICI)1097-0207(19970315)40:5<797::AID-NME89>3.0.CO;2-# |
| [33] | Gu, A boundary point interpolation method for stress analysis of solids, Computational Mechanics 28 pp 47– (2002) · Zbl 1115.74380 · doi:10.1007/s00466-001-0268-9 |
| [34] | Gu, A boundary radial point interpolation method (BRPIM) for 2-D structural analyses, Structural Engineering and Mechanics 15 (5) pp 535– (2003) · doi:10.12989/sem.2003.15.5.535 |
| [35] | Liu, An Introduction to Meshfree Methods and their Programming (2005) |
| [36] | Chen, Fractional diffusion equations by the Kansa method, Computers and Mathematics with Applications 59 (5) pp 1614– (2010) · Zbl 1189.35356 · doi:10.1016/j.camwa.2009.08.004 |
| [37] | Gu, An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation, CMES-Computer Modeling in Engineering and Sciences 56 (3) pp 303– (2010) · Zbl 1231.65178 |
| [38] | Zhu, A Modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method, Computational Mechanics 21 pp 211– (1998) · Zbl 0947.74080 · doi:10.1007/s004660050296 |
| [39] | Atluri, A critical assessment of the truly meshless local Petrov-Galerkin (MLPG), and local boundary integral equation (LBIE) methods, Computational Mechanics 24 pp 348– (1999) · Zbl 0977.74593 · doi:10.1007/s004660050457 |
| [40] | Belytschko, Smoothing and accelerated computations in the element free Galerkin method, Journal of Computational and Applied Mathematics 74 pp 111– (1996) · Zbl 0862.73058 · doi:10.1016/0377-0427(96)00020-9 |
| [41] | Liu, Mesh Free Methods: Moving Beyond the Finite Element Method (2009) · Zbl 1205.74003 · doi:10.1201/9781420082104 |
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