A high-order numerical scheme based on graded mesh and its analysis for the two-dimensional time-fractional convection-diffusion equation

Abstract

This work proposes a numerical method for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The time-fractional derivative is considered in the sense of Caputo. The considered problem with Caputo temporal fractional derivative generally shows a weak singularity at the initial time. In order to handle the weak singularity at t = 0 , the time-fractional derivative is discretized on a non-uniform graded mesh. A high-order compact operator is employed for approximating the derivatives in space directions. The resulting system of equations is then solved using the Alternating Direction Implicit (ADI) approach. We prove the stability and convergence of this scheme. Numerical experiment is performed to demonstrate the applicability and accuracy of the method. The considered numerical example subjected to nonsmooth solution confirms that the method has an order min ⁡ ( 2 − α , r α , 2 α + 1 ) in the temporal direction, where α ∈ ( 0 , 1 ) is the order of the fractional derivative and r is a parameter used in construction of the graded mesh and has a rate of convergence of order four in spatial direction. The proposed method on graded mesh has an advantage in terms of numerical accuracy over the method on the uniform mesh. The CPU time (in seconds) for the present method is also provided.