An efficient differential quadrature method for fractional advection-diffusion equation

TL;DR

This paper introduces a new differential quadrature method using cubic trigonometric B-splines for efficiently solving 1D and 2D fractional advection-diffusion equations, demonstrating high accuracy and stability through benchmark tests.

Contribution

It develops a novel DQ approach with explicit formulas for fractional derivatives, improving computational efficiency and accuracy for fractional ADEs.

Findings

Accurate solutions for five benchmark problems

Effective stability under mild conditions

Enhanced simulation of soliton and pulse propagation

Abstract

This article studies a direct numerical approach for fractional advection-diffusion equations (ADEs). Using a set of cubic trigonometric B-splines as test functions, a differential quadrature (DQ) method is firstly proposed for the 1D and 2D time-fractional ADEs of order $(0,1]$. The weighted coefficients are determined, and with them, the original equation is transformed into a group of general ordinary differential equations (ODEs), which are discretized by an effective difference scheme or Runge-Kutta method. The stability is investigated under a mild theoretical condition. Secondly, based on a set of cubic B-splines, we develop a new Crank-Nicolson type DQ method for the 2D space-fractional ADEs without advection. The DQ approximations to fractional derivatives are introduced and the values of the fractional derivatives of B-splines are computed by deriving explicit formulas. The presented DQ methods are evaluated on five benchmark problems and the concrete simulations of the unsteady propagation of solitons and Gaussian pulse. In comparison with the existing algorithms in the open literature, numerical results finally illustrate the validity and accuracy.