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A new difference scheme for the time fractional diffusion equation. (English) Zbl 1349.65261
Summary: In this paper we construct a new difference analog of the Caputo fractional derivative (called the $$L 2 - 1_\sigma$$ formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid $$L_2$$-norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by the numerical calculations carried out for some test problems.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35R11 Fractional partial differential equations 39A60 Applications of difference equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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