Finite Difference Method For Time Fractional Convection-diffusion Equation

Fractional calculus is a generalization of traditional integer calculus,Because fractional differential operators are non-local,they are suitable for describing materials with memory and genetic properties.However,analytical solutions of fractional differential equations are difficult to obtain,so it is particularly important to study numerical methods for solving fractional differential calculus.This paper studies two difference methods for one-and two-dimensional time-fractional convection-diffusion equations.In the time direction,the fractional derivative of the α-order Caputo time is discreted by the L1difference format.Firstly,this paper studies the central difference scheme of one-dimensional fractional convection-diffusion equations.In the spatial direction,the derivative term is discretized in a central difference format,the convergence order of the difference scheme is O(τ2-α+h2),where α(0<α<1)is the order of the fractional derivative,τ,h are the step sizes in time and space direction,it is proved in detail that the established difference format has unique solvability,stability and convergence when the space step satisfies |a(x)h|<2.Secondly,this paper the upwind difference scheme for one-dimensional fractional convection-diffusion equations.In the spatial direction,the diffusion term is approximated by the central difference scheme and the convection term is approximated by the upwind difference scheme,the convergence order of the difference scheme is O(τ2-α+h),the differential format established at this time satisfies the unique solvability,unconditional stability and convergence.Thirdly,this paper studies the central difference scheme of two-dimensional fractional convection-diffusion equations.In the spatial direction,the diffusion and convection terms are discretized using a central difference scheme,the convergence order of the difference scheme is O(τ2-α+h12+h22),where h1,h2 are the step sizes in the x,y directions,respectively.It is proved in detail that the established difference format has unique solvability,stability and convergence when the space step satisfies|a1h1|<2,|a2h2|<2.Finally,this paper studies the upwind difference scheme for two-dimensional fractional convection-diffusion equations.In the spatial direction,the diffusion term is approximated by the central difference scheme,and the convection term is approximated by the upwind difference scheme,the convergence order of the difference scheme is O(τ2-α+h1+h2).It also proves that the difference scheme has unique solvability,unconditional stability,and convergence.Numerical experiments verify the effectiveness of the two differential formats proposed in this paper.