Research On High Order Finite Difference Method For The Convection-diffusin Equation

In recent years,fractional differential equations have become the focus of research by experts and scholars at home and abroad due to their important practical application background.However,the non-locality of the fractional derivative operator results in that the analytical solutions of fractional differential equations usually require very complicated special series to represent.It is of great importance to research reasonable and feasible numerical algorithms of fractional differential equations.This dissertation mainly focuses on the unilateral space fractional convection diffusion equation,describing the abnormal diffusion of particles in a physical system.The fractional operator of this equation is defined by Riemann-Liouville(RL).In the existing literature,most of the RL fractional derivative was solved based on the first order classical and shifted r(5)(5)Letnikov-nwaldu G formula.And then,this dissertation derived a finite difference method with second order accuracy both in time and space based on linear spline interpolation.At the same time,we also attempted to use the smooth particle hydrodynamics(SPH)method,which is one of the meshless methods to numerically simulate the unilateral space fractional convection diffusion equation.The application of the SPH method in the fractional differential equation has not been involved before.The main work of this dissertation: Based on linear spline interpolation,a second order discretized scheme of the RL space fractional derivative combining with the Crank-Nicolson(CN)method on the time layer,the SCN method is presented for numerically solving the one-dimensional unilateral space fractional convection diffusion equation.The compatibility of the algorithm is analyzed,and the stability is proved by Fourier analysis method.The SCN method is theoretically proved to have second order accuracy both in time and space,and is unconditionally stable.Select numerical examples with analytical solutions,comparing the numerical results of SCN method and SPH method,and simulated the trend of examples without analytical solutions.Then these two methods have been extended to the two dimensional unilateral space fractional convection diffusion equation,and the alternating direction implicit scheme is used to solve the discrete scheme of the SCN method.