Fast Method For Space-time Fractional Partial Differential Equation And Its Application

Fractional calculus theory is a rapidly growing field of research in recent decades,which has been applied to describe complex phenomena in the mechanics and engineering,especially,the description of anomalous diffusion in complex sys-tems.Traditional diffusion represents the particles following a random walk,and this random walk limit is a Brownian motion.The anomalous diffusion can describe the probability density function of a particle following asymmetric den-sities,heavy tails,sharp peaks.Anomalous diffusion is generally observed and captured in massive real data.The object of these data might be pollutants in ground water,stock prices,sound waves,proteins crossing a cell boundary,or animals invading a new ecosystem.The anomalous transport regimes include generally sub-diffusion and super-diffusion.Fractional derivatives in space model super-diffusion,related to long power-law particle jumps.Fractional derivatives in time model sub-diffusion,related to long waiting times between particle jump-s.Hence,fractional model lies in the straightforward way of simulating more effectively and more accurately complex transport regimes.But,because of the character of history-dependence and the nonlocal property of fractional operators,it also increases the complexity of solving and simulating of fractional models.Since most of the fractional partial differential equations(FPDEs)cannot be analytically solved,the research of the FPDEs becomes important and necessary.Extensive research has been carried out in the development of numerical methods for FPDEs,including finite difference method[55-71];finite element method[72-96]and the spectral method[97-113],Other numerical methods,such as meshless methods[114-116];finite volume methods[117-119]and DG method[120,121].Because of the nonlocal property of fractional differential operators,the com-putational cost of solving fractional PDEs is much more expensive than that of solving integral-order PDEs.For solving the space-fractional PDEs,the stiffness matrices of above numerical methods are always full or dense.Traditional meth-ods to solve these numerical methods requires computational cost of O(N3)per time step and O(N2)of memory,where N is the number of spatial grid points.For solving the time-fractional PDEs,we need to store all the previous results in order to obtain the current numerical solution due to the history-dependence of the fractional operators.For solving the space-time fractional PDEs,numeri-cal methods for space-time FPDEs generate dense stiffness matrices and involve numerical solutions at all the previous time steps,and so have O(N2 + MN)memory requirement and O(MN3+M2N).computational complexity,where M is the numbers of time steps.The significantly increased computational complex-ity and memory requirement impose a serious challenge for numerical simulation of high dimensional fractional equations.The main contents of our thesis can be included as follow:In chapter 1,we introduce briefly the history and some definitions of fraction-al calculus.Then we recall the development of numerical methods for fractional PDEs,and introduce the Krylov subspace methods and it's pre-conditioners,Toeplitz matices and Circulant preconditioner.In chapter 2,we develop a fast finite difference method for time-dependent space-fractional PDEs with fractional derivative boundary conditions in three s-pace dimensions and prove the unconditional stability and convergence of the corresponding FDM.We have known that Wang et al.[122-126]accordingly de-veloped fast FDMs for FPDEs in one and multiple space dimensions.The stiff-ness matrices were proved to possess Toeplitz or block-Toeplitz-circulant-block(BTCB)like structures.This fast methods have approximately linear compu-tational complexity and linear storage per iteration when applying any Krylov subspace iterative method.But for the fractional derivative boundary-value prob-lem,the strong coupling of the boundary nodes in the two-dimensional boundary to the interior nodes in the three-dimensional physical domain significantly com-plicates the structure of the stiffness matrix of the FDM.For simplicity,we assume that the same number of meshes are used in the x,y,and z directions.Then the number of interior nodes is N and a fast matrix-vector multipliation of the interior nodes would be O(N log N).However,as the number of boundary nodes is O(N2/3),a naive matrix-vector multiplication of the boundary nodes would be O(NN2/3)= O(N5/3)that dominates the matrix-vector multiplication of the interior nodes! Therefore,a careful and thorough analysis has to be carried out in order to develop a fast FDM in the current context,which is the major task in this paper.In the numerical experiments,we give the constant coefficient,vari-able coefficient smooth solution and the constant coefficient non-smooth solution as the numerical experiments to verify the performance of the finite diifference scheme.In chapter 3,we develop and analyze fast numerical methods for space-time FPDEs in three space dimensions with a combination of Dirichlet and frac-tional Neumann boundary conditions.We discretize the Caputo time-fractional derivative by the L1 approximation and discretize space-fractional derivative by the shifted Grunwald finite difference scheme.Because of the nonlocal prop-erty of fractional differential operators,a time-marching FDS has an overall O(MN3 + M2N)computational complexity and O(N2 + MN)memory require-ment.We develop two global space-time fast method by carefully analyzing the structure of the stiffness matrix of the numerical discretization as well as the coupling in the time direction.Then we decelop a fast and efficient approx-imate method[134]which based on the sum-of-exponentials approximation for the convolution kernel t-1-? of time Caputo fractional derivative be applied to solve space-time FPDEs.The resulting numerical methods have almost linear computational complexity and linear storage with respect to the number of s-patial unknowns as well as time steps.Numerical experiments are presented to demonstrate the utility of the methods.In chapter 4,we develop a preconditioned fast Hermite finite element method for steady-state space-fractional diffusion equations,by proving that the stiffness matrix of the method can be expressed as a Toeplitz block matrix.Due to the stiffness matrix are ill-conditioned,the condition number of the stiffness matrix is huge with the increasing number of grids,even makes the iterative method di-vergent.So a block circulant preconditioner is presented.Numerical experiments are presented to demonstrate the utility of the methods.In chapter 5,the pore structure in shale reservoir shows strong heterogeneity.Nanopores shale gas includes free gas stored in pores and adsorbed gas in rock organic materials.The molecular diffusion rates of free gas and adsorbed gas have great difference.According to molecular dynamics(MD)modeling,the particle's mean square displacement(MSD)is characterized by a sublinear growth.In other word,The sublinear growth of MSD suggests that these transport processes are subdiffusive and the underlying particle motion can be described by a continuous time random walk,which yields an SDE driven by a time fractional PDE.MD simulation has provided a rigorous approach for modeling gas transport of shale gas at nanoscale,MD simulation can effectively estimate the transport diffusivity coefficient for pore space and rock space,but it can be computationally extreme-ly expensive that in turn restricts the application of MD simulations.Using an integrated fractional PDE and molecular dynamics(MD)modeling can fast and effectively study shale gas transport behavior in heterogeneity of nanopores structure.This new modeling can overcome the limitation of the extremely high computational cost of MD,what' s more,it is of crucial importance in efficiently recovering upscaled transport diffusivity coefficient of shale formations,which is of great significance to the economic development of shale gas reservoirs.