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

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.

65R20Integral equations (numerical methods)
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
45K05Integro-partial differential equations
35K55Nonlinear parabolic equations