Research On High Accuracy Difference Scheme Of Spatial Fractional Convection-diffusion Equation

Fractional calculus equations have evolved rapidly and are widely used in many fields in recent years.Fractional-order integral equations are more applicable than integer-order equations,but there are only a few fractional-order integral equations which analytical solutions can be obtained.As one of the most important differential equations in mathematical physics,the numerical solution of the fractional model of the convection-diffusion equation is of great significance.The thesis mainly studies the numerical solution of the spatial fractional convection-diffusion equation.The fractional derivatives in the article are all defined by Caputo fractional derivatives,where the order of the fractional term ? satisfies??(2,1].The Caputo fractional derivative terms are discretized by interpolation method,a high-precision difference scheme is established,and error analysis is performed.The specific content is as follows:Firstly,the Caputo fractional derivative is discretized by linear spline interpolation,and the first-order spatial derivative is discretized by central difference,combined with the Crank-Nicolson method,the difference scheme of the spatial fractional convection-diffusion equation with second-order convergence in both time and space directions is obtained.Secondly,the Richardson extrapolation method is used to further increase the accuracy,and specific algorithms are given,so that the new difference scheme of the convection-diffusion equation satisfies the second-order convergence in the time direction and the fourth-order convergence in the space direction.The fully discrete schemes of one-dimensional and two-dimensional convection-diffusion equations are given.The one-dimensional scheme is directly solved by Gauss elimination method,and the two-dimensional scheme cannot be solved directly,ADI method combined with Gauss elimination method is required for calculation.The error analysis is further performed,and the proof of the compatibility of the method is given,and the proof of the stability is given by the matrix method.Finally,this method is verified by numerical examples,and compared with other methods,the results show the effectiveness of the method.