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Tadjeran, Charles; Meerschaert, Mark M.; Scheffler, Hans-Peter

A second-order accurate numerical approximation for the fractional diffusion equation. (English) Zbl 1089.65089

J. Comput. Phys. 213, No. 1, 205-213 (2006).
Summary: 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.

Cited in 301 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
26A33 Fractional derivatives and integrals

Keywords:

Second-order accurate finite difference approximation; Stability; Crank-Nicholson method; algorithm; fractional order diffusion equations; consistency; convergence; numerical example
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\textit{C. Tadjeran} et al., J. Comput. Phys. 213, No. 1, 205--213 (2006; Zbl 1089.65089)
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