A second-order accurate numerical approximation for the fractional diffusion equation
Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, we examine a practical numerical method which is second-order accurate in time and in space to solve a class of initial-boundary value fractional diffusi...
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| Veröffentlicht in: | Journal of computational physics 2006-03, Vol.213 (1), p.205-213 |
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| container_title | Journal of computational physics |
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| creator | Tadjeran, Charles Meerschaert, Mark M. Scheffler, Hans-Peter |
| description | Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, we examine a practical numerical method which is second-order accurate in time and in space to solve a class of initial-boundary value fractional diffusive equations with variable coefficients on a finite domain. An approach based on the classical Crank–Nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Stability, consistency, and (therefore) convergence of the method are examined. It is shown that the fractional Crank–Nicholson method based on the shifted Grünwald formula is unconditionally stable. A numerical example is presented and compared with the exact analytical solution for its order of convergence. |
| doi_str_mv | 10.1016/j.jcp.2005.08.008 |
| format | Article |
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| ispartof | Journal of computational physics, 2006-03, Vol.213 (1), p.205-213 |
| issn | 0021-9991 1090-2716 |
| language | eng |
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| source | Elsevier ScienceDirect Journals |
| subjects | Approximation Computational techniques Consistency Convergence Crank-Nicholson method Diffusion Exact sciences and technology Extrapolation Fractional partial differential equation Mathematical analysis Mathematical methods in physics Mathematical models Numerical algorithm for superdiffusion Numerical fractional PDE Physics Second-order accurate finite difference approximation Stability analysis |
| title | A second-order accurate numerical approximation for the fractional diffusion equation |
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