Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

Abstract

In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems

“…Table 1 shows the effect of applying uniform and non-uniform PDQM on computation of solute concentration υ(x, t) at β = 1.85, C x = 2, L = 8. It is noticed that the L ∞ error is decreasing when the grid points are increasing, that is, the PDQM is stability in the x-direction, as well as the computed results via non-uniform grids are higher agree with earlier numerical solutions [13,42,43,59] than uniform ones. Also, L ∞ error (1.0885 × 10 −6 ) and performance time (0.17 s) for non-uniform PDQM achieved the least.…”

“…The primary objective of our paper is to evaluate the performance, validity, efficiency, and accuracy of the developed techniques. We confirm this by comparing the computed results with existing numerical solutions [13,42,43,59] and analytical solutions [40,41,44]. To assess the convergence and accuracy of the developed methods, we use the error computation method outlined in [13,42,43,59]:…”

“…We confirm this by comparing the computed results with existing numerical solutions [13,42,43,59] and analytical solutions [40,41,44]. To assess the convergence and accuracy of the developed methods, we use the error computation method outlined in [13,42,43,59]:…”

“…Additionally, we design a MATLAB code for each approach to obtain a numerical solution for three problems to be examined. When compared to earlier analytical [40,41] and numerical [13,42,43] methodologies, the derived numerical results attain excellent efficiency and accuracy. Furthermore, we present some parametric studies to highlight the reliability of our methods with the influence of fractional order derivative, the velocity, and positive diffusion parameters on the results.…”

An efficient numerical scheme and its analysis for the multiterm time‐fractional convection‐diffusion‐reaction equation

“…Li and Wang [25] proposed a method based on graded mesh in time and local discontinuous Galerkin finite element method in space to solve the TFCDR equation with nonsmooth solution. Dong et al [26] considered the 2D time‐fractional convecton‐diffusion equation with order of fractional derivative $$$ \alpha \in \left(1,2\right) $$$. They proposed a compact ADI scheme to solve this problem.…”