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Partially Explicit Time Discretization for Nonlinear Time Fractional Diffusion Equations

Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there are high contrast changes in media properties. For solving these problems, one often uses coarse spatial grid for spatial resolution. For temporal discretization, implicit methods are often used. For implicit methods, though the time step can be relatively large, the equations are difficult to compute due to the nonlinearity and the fact that one deals with large-scale systems. On the other hand, the discrete system in explicit methods are easier to compute but it requires small time steps. In this work, we propose the partially explicit scheme following earlier works on developing partially explicit methods for nonlinear diffusion equations. In this scheme, the diffusion term is treated partially explicitly and the reaction term is treated fully explicitly. With the appropriate construction of spaces and stability analysis, we find that the required time step in our proposed scheme scales as the coarse mesh size, which creates a great saving in computing. The main novelty of this work is the extension of our earlier works for diffusion equations to time fractional diffusion equations. For the case of fractional diffusion equations, the constraints on time steps are more severe and the proposed methods alleviate this since the time step in partially explicit method scales as the coarse mesh size. We present stability results. Numerical results are presented where we compare our proposed partially explicit methods with a fully implicit approach. We show that our proposed approach provides similar results, while treating many degrees of freedom in nonlinear terms explicitly.