Numerical Simulations And Analysis For Time-dependent Space Fractional Diffusion Equations

In recent decades,fractional partial differential equations(FPDEs)are wide-ly used in complex systems,which are in the field of engineering and science.In terms of integer-order derivatives,fractional derivatives are more suitable to describe some complex environments,which have the properties of heredity or memorability,long range dependence,and non-locality.Thus FPDEs become a powerful tool for modeling complex phenomena,such as anomalous diffusion,pollutant transport,random dynamic system,finance,risk estimates and so on.However,FPDEs involve in complex and singular integral operators.It is chal-lenging to search very effective numerical methods to solve these equations.In addition,considering the fractional operators' non-locality,the corresponding nu-merical methods would generate dense or full stiffness matrices.So it is of practi-cal significance to find fast iterative algorithms to solve linear algebraic systems.Two types of time-dependent space fractional diffusion equations are studied in this dissertation,They are fractional Fokker-Planck advection diffusion equation and fractional Gray-Scott reaction diffusion equations.Based on this two FPDEs,this dissertation mainly investigates theoretical analysis,numerical methods,er-ror estimates,numerical simulations or comparison and data analysis.The details are:In chapter 2,firstly,we introduce a one-dimensional space-fractional advec-tion diffusion equation,which describes a super-diffusion transport phenomenon.pt+(V(x,t)p)x-dpxx+?(-?)?/2p=f(x,t),x?R,t?(0,T],with Under the homogeneous boundary condition and initial condition,we use the Eulerian-Lagrangian localized adjoint method(ELLAM)to deduce the numer-ical scheme.This method exhibits the advantages of alleviating the Courant number restrictions,reducing the time truncation errors and maintaining mass conservation.They can generate accurate numerical solutions even if the mesh is coarse and the time step is large.We make error estimates for this numerical scheme and prove that when the exact solution p ? L?(0,T;Hs)? H1(0,T;Hs),this scheme is of order O(?t+hs+hs+1-?-?).Considering the non-locality of the fractional Laplacian,numerical scheme would generate a dense or full stiff-ness matrices.We study the structures of the stiffness matrices and prove they possess tridiagonal and Toeplitz structures.Thus,by using the fast Fourier trans-form and fast matrix-vector multiplication,we obtain a fast conjugate gradient method,which could reduce computational complexity from O(I3)by traditional iterative algorithm to O(Ilog2I)and reduce memory from O(I2)to O(I),and loss no accuracy.Two numerical experiments verify the accuracy and efficien-cy of this numerical method.The solutions show that when the mesh is coarse and time step is large,ELLAM is more accurate than the Backward-Eulerian scheme.It also can obtain second-order rate of convergence in space.Comparing with traditional Gaussian elimination method and conjugate gradient method,fast conjugate gradient method losses no accuracy.For linear algebraic system of order 512,the CPU time consumption is more than 4 hours by Gaussian e-limination method,while the fast conjugate gradient method just consumes 14 seconds.In chapter 3,we first show the two-dimensional nonlinear space-fractional Gray-Scott(GS)model on rectangle domain with Then we solve the uniform steady states of this model and analysis the linear stabilities of these steady states.The GS model has three steady states,and the stabilities of steady states vary with the values of parameters F and ?.We obtain the value range of the two parameters,which provides theoretical basis for choosing parameter values in numerical simulations.Considering the com-plexity of the GS model,we prove its well-posedness.Under the homogeneous boundary condition and initial condition,we use finite difference method to solve this model,i.e.,using Crank-Nicolson(C-N)scheme for time discretization and using weighted shifted Grunwald difference operators to approximate fractional derivatives.We use the second-order implicit-explicit methods to handle the non-linear terms.We provide the stability analysis for the time-discrete scheme.In numerical experiments,we use a benchmark problem to test the accuracy of the numerical scheme.The numerical solutions exhibit that this numerical scheme has second-order accuracy in time and space.By using a initial perturbation to the spatially homogeneous steady state,we observe the formation of patterns under the condition of different fractional order ?,? and parameter values.For?=?,when we choose different values of F and ?,pattern formation propagates outward by mitosis and circular wave.When ??? under the same condition of F and ?,the formation of patterns changes.Since it is difficult to obtain the exact solutions of the fractional GS model,we use spectral collocation method in space discretization to solve this problem and compare the steady patterns obtained by the two different methods.Further more,we also calculate the cor-responding radial distribution functions(RDFs)for comparison.The comparison results show that the numerical results obtained by two different numerical meth-ods are almost the same.The corresponding RDFs further verify this point.The comparison results demonstrates that the second-order implicit-explicit finite d-ifference method is accuracy.Finally,we estimate the scaling law between the fractional orders and the distance between all spot pairs in steady spot patterns,which is significant in the analysis of molecular dynamics.Since the diffusion dominated by Riesz fractional derivatives propagates a-long directions of coordinate axis,we generalize the GS model by using the inte-gral of fractional directional derivatives as fractional diffusion operators,i.e.,We know that,once the angle ? is specified,fractional directional derivatives could reduce to Riemann-Liouville fractional derivatives.We do some numerical research on the generalized GS model.We discretize this fractional diffusion equa-tions in space by using fast finite element method.By estimating the fractional directional derivatives of the basis functions,we study that the stiffness matrices of the numerical scheme possess block-Toeplitz-Toeplitz-block structures.Thus a fast matrix-vector multiplication can applied to iterative algorithms.