A second-order accurate numerical method for the two-dimensional fractional diffusion equation. (English) Zbl 1113.65124

Tadjeran, Charles; Meerschaert, Mark M.

J. Comput. Phys. 220, No. 2, 813-823 (2007).

The paper is devoted to a study of a numerical method for space-fractional diffusion equations in two space dimensions, with the differential operators being of Riemann-Liouville type. In contrast to the common practice, the authors impose classical Dirichlet boundary conditions. Moreover they always assume that the unknown solutions are sufficiently well behaved for the analysis of the numerical scheme to go through, without discussing concrete conditions under which this assumption is satisfied.
The algorithm itself is based on a shifted Grünwald approximation of the fractional space derivatives, combined in an alternating direction implicit fashion with a Crank-Nicolson approach for the first-order time derivative. The result is (under the unspecified conditions) an unconditionally stable method with second order accuracy in time and first order accuracy in space. By adding a Richardson extrapolation step, the accuracy with respect to space can be improved to second order as well.

Reviewer: Kai Diethelm (Braunschweig)

MSC:

65R20	Numerical methods for integral equations
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
45K05	Integro-partial differential equations
35K55	Nonlinear parabolic equations

Keywords:

space-fractional diffusion equation; extrapolation; Crank-Nicolson method; alternating directions implicit method; ADI method; Grünwald formula; algorithm

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