Novel numerical methods based on graded, adaptive and uniform meshes for a time-fractional advection-diffusion equation subjected to weakly singular solution

This paper presents adaptive, graded and uniform mesh schemes to approximate the solution of a fractional order advection-diffusion model, which generally shows a weak singularity at the initial time level. The temporal fractional derivative in the underlying problem is described in a Caputo form and is discretized by means of L1 scheme on a nonuniform mesh. The space derivative is discretized on a uniform mesh employing a fourth-order compact finite difference scheme. The adaptive grid is generated via equidistribution of a positive monitor function. Stability and convergence results for the proposed method on graded mesh are established. Numerical examples are provided to study the accuracy and efficiency of the proposed techniques and to support the theoretical results. A discussion about the advantages of the graded and adaptive meshes over the uniform one is also presented. The CPU times for the proposed numerical schemes are provided.