Lattice Boltzmann Modeling And Numerical Simulation For Two Kinds Of Time-fractional Differential Equations

In recent years,the rapid development of fractional calculus promotes its application in the fields of anomalous diffusion,turbulence,viscoelastic mechanics and so on.Fractional calcu-lus can describe many physical processes more accurately since its nonlocal property.In this paper,the lattice Boltzmann models with BGK operator for solving two kinds of time fractional differential equations are proposed based on the fast evalution of Caputo derivative.The cor-rectness and effectiveness of these models are verified through Chapman-Enskog analysis and abundant numerical examples.For the time fractional wave equation,the Caputo derivative is discretized firstly after or-der reduction,which is divided into the historical part and the current part.The former part is approximated using the efficient algorithm for the evalution of the Caputo fractional deriva-tive,while the latter is simply approximated by rectangle formula to keep the time-dependent characteristics of wave equation.Then,the lattice Boltzmann-BGK model is constructed by means of velocity discretization,time and space discretization,selecting appropriate equilib-rium distribution function and force distribution function.Through Chapman-Enskog analysis,the corresponding macroscopic equation is derived and the correctness and validity of the model are proved.Besides,the relationship between the relaxation time?and the parameter in macro-scopic equation is derived in this process.Numerical examples are given to further demonstrate the effectiveness of this model,and second order convergence in space is verified from numer-ical results.Based on the lattice Boltzmann model for time fractional wave equation,the lattice Boltz-mann models for time fractional Klein-Gordon equation and Sine-Gordon equation are proposed and are used to solve the initial-boundary problems with exact solutions.The results show that the numerical solution of lattice Boltzmann model is in good agreement with the analytical solu-tion.Furthermore,the lattice Boltzmann model is used to simulate the evolution of circular ring solitons which is described by the time fractional Sine-Gordon equation.The lattice Boltzmann numerical results are in good agreement with the existing literatures.In addition,the influence of the order of Caputo fractional derivative on the evolution of solitons is also simulated using this model.Finally,by using the above discretization method of Caputo fractional derivatives,a lattice Boltzmann-BGK model for solving time-fractional incompressible Navier-Stokes equations is proposed.The correctness and validity of this model are proved theoretically through Chapman-Enskog analysis.Compared with the above model,the equilibrium distribution function and external force of this model are more complex because fractional Navier-Stokes equations are coupled systems and contain nonlinear convection term.Consequently the process of Chapman-Enskog analysis is more complicated especially in the calculation of?_??.In addition,since the derivation of Navier-Stokes equations from lattice Boltzmann equation requires the equilibrium distribution function to satisfy higher-order moment conditions,a more symmetric lattice veloc-ity set must be used.In the numerical experiment,a two-dimensional problem with analytical solution is selected for numerical verification.The effectiveness of the model is verified by comparing the maximum errors between the numerical solution and the analytical solution.