Study Of Numerical Methods For Surface Fractional Convection Diffusion Equations

Fractional partial differential equations refer to the equations with fraction-al power in unknown variables.It is more suitable than traditional integral-order equations for describing real-world practical problems with the memory and hered-ity of various materials,such as electrolytic chemistry,condensed matter physics,semiconductor physics,turbulence and viscoelasticity systems,biomathematics and statistical mechanics,optics and heat systems,materials and signal processing.Frac-tional convection diffusion problem is the most widely used problem in scientific and engineering computational simulation,many practical fluids flow processes,such as heat transfer,fluid mechanics,groundwater pollution transport and diffusion pro-cesses and reservoirs,mass and energy transfer,and global weather simulation,can all be described by fractional convection diffusion models.But most of these prob-lems have no exact solution at present,and an accurate and reliable discrete format is required for numerical solution.The numerical solution of the surface fractional convection diffusion equation can be directly used to simulate the fluid motion on the surface.Based on its application background in fluid mechanics,physics,ma-terials science and biology,the research on the numerical method of the fractional convection diffusion problem on surfaces is of great significance to both theory and practice.At present,the more numerical methods used are finite difference method and finite element method.However,due to the computational difficulties caused by the nonlocality of fractional order,most of the problems stay in the study of low-dimensional and simple regions,and there is not much work on the research of high-dimensional and complex region problems.In this paper,several stable and efficient numerical methods are constructed for the surface fractional convection d-iffusion model,and the corresponding numerical schemes are analyzed theoretically The specific contents are as followsFirstly,the application and numerical algorithm of time fractional convection d-iffusion equation are studied.The radial basis function difference method is applied,and we use the compact support Wendland radial basis function with a polynomial to determine the weight of adjacent points around the center point of spatial deriva-tive.In time discretization,we use the shifted Gr(?)nwald formula,which can obtain the second-order accuracy in space and time.Moreover,we provide an adaptive s-trategy to select the support region and get an effective shape parameter.The main idea is to consider the distribution of the center point in the support domain so that the number of neighbor points around the center point is about a constant.In this way,a coefficient matrix with good symmetry and sparsity can be obtained.In addition,the stability and convergence of the proposed method are proved.Finally,the method is tested the accuracy of the method by numerical experiments.The method is a good solution to the ill-posed problem caused by the global radial basis function method,and at the same time improves the problem of the lower order of the radial basis function finite difference methodSecond,the local compact integral radial basis function is used for the time fractional convection diffusion reaction equation.The scheme combines the inte-gral radial basis function approximation and compact approximation,uses the node function value and second derivative value to establish the relationship between the physical space and radial basis function weight space,and discretizes the spatial derivative according to the information of adjacent points in the template.More-over,the stability of the proposed method are deduced.Some numerical examples on 2D and 3D domains are carried out to verify the accuracy and fast convergence speed of the proposed algorithm.This method optimizes the local radial basis func-tion method to keep the advantage of the coefficient matrix condition number small,while improving the accuracy of the local radial basis function methodThirdly,the application of radial kernel to time fractional convection diffusion equation on R3 closed surface.All computations use only extrinsic coordinates to avoid coordinate distortions and singularities.The non-locality of the fractional or-der makes it difficult to solve the model calculation.Because it not only depends on the current state,but also largely depends on all past states.In addition,the surface problem requires a large amount of calculation,which is a challenge in storage and calculation efficiency.To solve this problem,radial kernel function approximation method is employed in space discretization and the second order shifted Gr(?)nwald scheme is applied in time discretization.Stability and convergency of the method are proven by the energy estimate.Finally,the numerical experiments and appli-cations of the time fractional convection diffusion PDEs on the surface are carried out.This method makes progress in numerical calculation of the surface fractional convection diffusion equation.Fourth,the problem of space fractional reaction-diffusion equation on closed surface is discussed.The definitions of space fractional Laplace-Beltrami operator(-?M)?/2,??(1,2]on surfaces are given,and the space discretization is imple-mented by the surface finite element method.Applying the matrix transfer technique for the space fractional,the fractional Laplace-Beltrami operator is approximated as A?/2u,and A is the sparse approximation of-?M which can be expressed diagonally by eigenvalue decomposition,Which subtly solves the solution difficulty caused by spatial fractional order non-locality.In addition,a large number of numerical ex-periments on different surfaces show that the method is feasible.In this method,fractional Laplace Beltrami operator on the surface is proposed for the first time,and combined with the existing methods,the surface space fractional reaction-diffusion equation is well applied.