A Fast Characteristic Finite Difference Method For The Distributed-Order Advection-Diffusion Equations

The main contents of this paper are as follows:a characteristic finite difference scheme and a fast calculation method of the spatial distribution order convection-diffusion equation.The model is as follows:(?)where P(?)is Non-negative weight functionand,and satisfies the following conditions:O ? P(?),P(?)?0,? ?[1,2],0<?12 P(?)<?,where V(x,t)is the average velocity of the fluid,K is a diffusion coefficient and K>0,f(x,t)is the source sink term.(?)?u/(?)|x|?/(x,t)represents Riesz fractional derivative operator,and defined as follows:(?)where(?)?u(x,t)/(?)+x? and(?)?u(x,t)/(?)-x?,(n-1<?<n)represent left and right ? order Riemann-Liouville fractional derivative respectively,and the specific expres-sion is:(?)where ?(·)is Gamma function.In this paper,the characteristic line method is used to transform the distributed convection-diffusion equation(1.1)iinto the distributed diffusion equation.And discrete the order of the distribution??time and space,then the characteristic finite difference scheme is obtained.Then the stability and error of the difference scheme are analyzed,and the format is found to be unconditionally stable.The space and time are first order convergence,distribution order is second order convergence.In addition,because time fract.ional derivative has non-local property,it is necessary to use the numerical results of all previous time layers to solve the current time layer.Then,the coefficient matrix of the discrete difference scheme is a full matrix.The amount of computation and storage will increase greatly as the mesh is encrypted.As a result,the calculation time becomes longer.It can be found by analyzing the structure of the coefficient matrix corresponding to the discrete equation,the coefficients matrix is Toeplitz matrix.Using fast Fourier transform and its inverse,the matrix can be stored efficiently.On this basis,we propose a fast conjugate gradient method based on fast matrix-vector multiplication and cyclic pretreatment.Numerical experiments show that this method has strong application potential,and greatly reduced CPU utilization.In this paper,we mainly discuss the characteristic difference scheme,theoretical analysis and corresponding fast algorithm for the above spatial distribution order model.This paper is divided into four chapters:Chapter 1:The background of the space distributed-order equations,the model described as above and domestic and foreign research literature review are introduced.Chapter 2:The characteristic finite difference scheme of convection-diffusion equations with spatial order is given,and discuss the convergence and stability of the scheme.Chapter 3:A fast algorithm of characteristic finite difference scheme is given,and numerical examples are given to verify the convergence order of the difference scheme and the effectiveness of the fast algorithm.Chapter 4:A summary and outlook of this paper are given.