Efficient numerical algorithms for solving a time-fractional diffusion equation with weakly singular solution

This manuscript presents efficient numerical techniques based on L1 approximation to handle the initial singularity of a time-fractional diffusion (TFD) problem. In this scheme, the temporal derivative is discretized on a non-uniform grid, while the spatial derivative is discretized on a uniform grid using a compact finite difference method (CFDM). In the temporal direction, we have considered two different non-uniform meshes, namely, the graded mesh and the adaptive moving mesh. In the case of adaptive mesh, the mesh is generated adaptively by equidistribution of a positive monitor function. The unconditional stability of the numerical scheme based on graded mesh is discussed. Convergence analysis of the method based on graded mesh is investigated. The advantage of the proposed adaptive mesh algorithm is examined by comparing the numerical accuracy to that of the standard uniform-mesh method and graded mesh method (Stynes et al., 2017).