Partially Explicit Time Discretization for Time Fractional Diffusion Equation

TL;DR

This paper introduces a partially explicit time discretization method for time fractional PDEs that allows larger time steps independent of media heterogeneity, improving computational efficiency in multiscale applications.

Contribution

The paper proposes a novel partial explicit scheme for time fractional PDEs that achieves stability with larger time steps, independent of media contrast or spatial scale.

Findings

The method is stable under the proposed conditions.

Numerical results confirm the stability and efficiency.

The approach allows larger time steps than traditional explicit methods.

Abstract

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 computations. 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.