Time fractional PDEs have been used in many applications for modeling and simulations. Many of these applications are multiscale and contain high contrast variations in the media properties. It requires very small time step size to perform detailed c
omputations. On the other hand, in the presence of small spatial grids, very small time step size is required for explicit methods. Explicit methods have many advantages as we discuss in the paper. In this paper, we propose a partial explicit method for time fractional PDEs. The approach solves the forward problem on a coarse computational grid, which is much larger than spatial heterogeneities, and requires only a few degrees of freedom to be treated implicitly. Via the construction of appropriate spaces and careful stability analysis, we can show that the time step can be chosen not to depend on the contrast or scale as the coarse mesh size. Thus, one can use larger time step size in an explicit approach. We present stability theory for our proposed method and our numerical results confirm the stability findings and demonstrate the performance of the approach.
In the current work we build a difference analog of the Caputo fractional derivative with generalized memory kernel ($_lambda$L2-1$_sigma$ formula). The fundamental features of this difference operator are studied and on its ground some difference sc
hemes generating approximations of the second order in time for the generalized time-fractional diffusion equation with variable coefficients are worked out. We have proved stability and convergence of the given schemes in the grid $L_2$ - norm with the rate equal to the order of the approximation error. The achieved results are supported by the numerical computations performed for some test problems.
