Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation

Abstract

a b s t r a c tIn this paper, we consider a two-sided space-fractional diffusion equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new fractional finite volume method for the two-sided space-fractional diffusion equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.

A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh

“…Previous studies showed that finite volume methods are anticipated to have a second-order convergence rate with respect to the mesh size on a uniform mesh [9,13,20,27,28] without using a Richardson extrapolation. This is in contrast to the Meerschaert-Tadjeran finite difference scheme, in which a Richardson extrapolation is required to recover a second-order accuracy [18].…”

“…Extensive research has been conducted in the development of numerical methods for fractional differential equations [5,7,9,10,[12][13][14]18,[26][27][28]. Because of the nonlocal nature of fractional differential operators, numerical methods for space-fractional differential equations usually generate full stiffness matrices and were traditionally solved via Gaussian elimination.…”

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

“…Khan et al [18] derived the exact solution for the accelerating flow of a generalized Oldroyd-B equation by utilizing the fractional calculus approach and discrete Laplace transform. Feng et al [19] studied the stability of a two-sided space-fractional diffusion equation. Liu et al have proposed some numerical methods for related fractional partial differential equations [20][21][22][23][24][25][26][27].…”

A granular computing method for nonlinear convection-diffusion equation

“…Therefore, numerical methods must be applied. At present, the major numerical methods are finite element methods(FEM) [2][3][4] [5], finite difference methods (FDM) [6] [7], finite volume methods(FVM) [8][9] [10]. However, when the FEM or FDM are used to solve the NCDE, it exhibits excessive numerical diffusion and nonphysical oscillation [11].…”