A robust higher-order numerical technique with graded and harmonic meshes for the time-fractional diffusion–advection–reaction equation

The solution of a time-fractional differential equation usually exhibits a weak singularity near the initial time t=0. It causes traditional numerical methods with uniform mesh to typically lose their accuracy. The technique of nonuniform mesh based on reasonable regularity of the solution has been found to be a very effective way to recover the accuracy. The current work aims to devise a compact finite difference scheme with temporal graded and harmonic meshes for solving time-fractional diffusion–advection–reaction equations with non-smooth solutions. The time-fractional operator involved in this is taken in the Caputo sense. A theoretical analysis of the stability and convergence of the proposed numerical technique is presented by von Neumann’s method. The scheme’s proficiency, robustness, and effectiveness are examined through three numerical experiments. A comparison of the numerical results on the uniform, graded, and harmonic meshes is presented to demonstrate the advantage of the suggested meshes over the uniform mesh. The tabular and graphical representations of numerical results confirm the high accuracy and versatility of the scheme.