High‐order schemes and their error analysis for generalized variable coefficients fractional reaction–diffusion equations

Abstract

In this manuscript, we develop and analyze two high‐order schemes, CFD and PQS , for generalized variable coefficients fractional reaction–diffusion equations. The generalized fractional derivative is characterized by a weight function and a scale function. We approximate it using generalized Alikhanov formula ( ) of order , where denotes the order of the generalized fractional derivative. Moreover, for spatial discretization, we use a compact operator in CFD scheme and parametric quintic splines in PQS scheme. The stability and convergence analysis of both schemes are demonstrated thoroughly using the discrete energy method in the ‐norm. It is shown that the convergence orders of the CFD and PQS schemes are and , respectively, where and represent the mesh spacing in the time direction and is the mesh spacing in the space direction. In addition, numerical results are obtained for three test problems to validate the theory and demonstrate the efficiency and superiority of the proposed schemes.

A Note on the Convergence of Multigrid Methods for the Riesz–Space Equation and an Application to Image Deblurring

“…We study a one-dimensional RFDE discretized with a numerical scheme with precision order one mainly for the simplicity of the presentation. Nevertheless, the same analysis can be carried out either when using higher-order discretization schemes [1,2] or when dealing with more complex higher-dimensional fractional models [3,4].…”

“…The robustness of the geometric multigrid can be improved by using it as a preconditioner, as seen in the column PV 1 1 . In particular, it is even more robust than the band multigrid preconditioner P s V 1 1 . When comparing the circulant preconditioners, the Strang preconditioner P S is preferable to the optimal Chan preconditioner P C .…”