2023/08/17 by Singh, Anshima, Kumar, Sunil
#65M06 #65M12 #65M70 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2308.08971
This study presents a novel high-order numerical method designed for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The Caputo definition is employed to characterize the time-fractional derivative. A weak singularity at the initial time (t=0) is encountered in the considered problem, which is effectively managed by adopting a discretization approach for the time-fractional derivative, where Alikhanov's high-order L2-1σ formula is applied on a non-uniform fitted mesh, resulting in successful tackling of the singularity. A high-order two-dimensional compact operator is implemented to approximate the spatial variables. The alternating direction implicit (ADI) approach is then employed to solve the resulting system of equations by decomposing the two-dimensional problem into two separate one-dimensional problems. The theoretical analysis, encompassing both stability and convergence aspects, has been conducted comprehensively, and it has shown that method is convergent with an order \mathcal O(Nt^-min\3-α,θα,1+2α,2+α\+hx4+hy4), where α∈(0,1) represents the order of the fractional derivative, Nt is the temporal discretization parameter and hx and hy represent spatial mesh widths. Moreover, the parameter θ is utilized in the construction of the fitted mesh.