  The present paper is devoted to constructing L2 type difference analog of the
Caputo fractional derivative. The fundamental features of this difference
operator are studied and it is used to construct difference schemes generating
approximations of the second and fourth order in space and the $(3-\alpha)$
th-order in time for the time fractional diffusion equation with variable
coefficients. Stability of the schemes under consideration as well as their
convergence with the rate equal to the order of the approximation error are
proven. The received results are supported by the numerical computations
performed for some test problems.
