Numerical Investigation Of Time/Space Fractional Convection Diffusion Equation Based On Corrected SPH Method

Because of the long-term memory and nonlocal properties of fractional derivatives,the nonlinear dynamic behavior of viscoelastic materials or rheology with memory and inheritance properties can be described more accurately.The nonlinear behavior is usually described by the time or space fractional convection-diffusion equation.However,the complexity of the fractional derivative or the nonlinear part makes it difficult to obtain the theoretical solution of the fractional convection-diffusion equation by analytical means in many cases.Therefore,scholars began to pay extensive attention to the numerical methods of fractional convection-diffusion equations,such as the finite difference method and finite element method of mesh class.However,the numerical method based on grid has many difficulties when dealing with complex irregular region problems or implementing local encryption.Therefore,in recent years,the pure meshless method-smooth particle hydrodynamics(SPH)method,which is completely independent of mesh,with its arbitrary points or the advantages of easy to deal with irregular area attention widely in the field of computational mechanics have become a new kind of calculation method,and the convection diffusion equation in solving fractional order also did not see related literature reports on the issue.When the traditional SPH method is directly applied to the solution of fractional-order convection-diffusion problem,it has the defects of low precision and poor stability.Therefore,it is necessary to further modify the traditional SPH method and develop a stable and accurate pure meshless algorithm.Based on the above analysis,the first derivative kernel gradient corrected SPH(CSPH)method is coupled with Caputo time fractional-order difference scheme to numerically investigate the constant/variable time fractional-order convection-diffusion equation.Secondly,by combining the integral discrete scheme of Riemann-Liouville fractional derivative with the CSPH scheme,a pure meshless discrete scheme(CSPH-SFCD),which can accurately solve the time-dependent space fractional convection-diffusion equation,is derived for the first time,and the numerical simulation of the spatial fractional Burgers equation is carried out by using it.The main research contents of this paper are as follows:1)Coupling the CSPH method with Caputo Discrete Difference Scheme based on time fractional derivative,a pure meshless CSPH-FDM method which can accurately solve the constant/variable TF-CDE is presented.In order to verify the numerical convergence order of CSPH-FDM discrete scheme,the errors of one dimensional or two dimensional constant/variable TF-CDE with analytic solution with Neumann boundary are analyzed.In order to demonstrate its flexible application,the simulation results of local encryption and complex irregular region problems are discussed.Numerical results show that the pure meshless method has better second-order accuracy and flexible generalization.2)CSPH-FDM is used to simulate the non-analytical solution TF-CDE,and compared with other results,the process of solute evolution with time is predicted successfully.3)By combining an integral scheme with CSPH formula,the Riemann-Liouville-based single and bilateral space fractional derivatives are discretized,and a grid-independent CSPH-SFCD discretization method for accurate solution of time-dependent SF-CDE equation is presented for the first time.The numerical convergence of one dimensional/two dimensional space fractional convection-diffusion equation and its analytical solution is analyzed.The flexible application of the pure meshless method in the case of non-rectangular region or local encryption is also discussed.4)The CSPH-SFCD is coupled with the upwind scheme to simulate and predict the nonlinear spatial fractional Burgers problem,and the results are compared with the finite difference method.The numerical results show that the pure meshless method proposed in this paper is effective and trustworthy to simulate complex SF-CDE problems.