Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients

Abstract

Space non-integer order convection–diffusion descriptions are generalized form of integer order convection–diffusion problems expressing super diffusive and convective transport processes. In this article, we propose finite difference approximation for space fractional convection–diffusion model having space variable coefficients on the given bounded domain over time and space. It is shown that the Crank–Nicolson difference scheme based on the right shifted Grünwald–Letnikov difference formula is unconditionally stable and it is also of second order consistency both in temporal and spatial terms with extrapolation to the limit approach. Numerical experiments are tested to verify the efficiency of our theoretical analysis and confirm order of convergence.

Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

“…An improved matrix transform numerical method is proposed in Reference [23] to solve one dimensional space fractional advection-dispersion model and its analytical solution is found using padé approximation. Recently, space fractional convection-diffusion with variable coefficients are solved using shifted Grünwald-Letnikov difference operator for space and Crank-Nicolson scheme for time that produce second order convergence both in time and space with extrapolation was studied [24].…”

A research study on unsteady state convection diffusion flow with adoption of the finite volume technique

“…tion (9) for different values of time t, then for the various nodes we have discretized equations. These simultaneous linear equations are evaluated in MATLAB [20] using the successive-over relaxation (SOR) method, and then temperature profile values for all nodes (x) are represented in Table 3.…”

“…Such an equation has an analytic solution are only possible for a restricted and limited number of cases [7]. Due to this, several numerical techniques have been developed for the numerical approximations of convection-diffusion problems [8,9]. The finite difference approximation method, the finite element method, as well as finite volume techniques are most widely used for computational fluid dynamics (CFD) [10,11].…”

Anomalous Solute Transport in a Cylindrical Two-Zone Medium with Fractal Structure

“…Equations (32) and (33) are solved using the one-dimensional Thomas' algorithm and are given the following form…”

“…In [32], finite-difference approximation was proposed for the space fractional convective-diffusion model, which has the coefficients of space variable on the given bounded domain over time and space. It was shown that the Crank-Nicolson difference scheme based on the right shifted Grünwald-Letnikov difference formula is unconditionally stable and also has second-order consistency both in temporal and spatial terms with extrapolation to the limit approach.…”