Fast Finite Difference/element Methods For Space-time Fractional Differential Equations

Fractional partial differential equations(PDEs)were shown to provide a more powerful means for modeling challenging phenomena such as anoma-lously diffusive transport or long-range spatial interactions and memory effec-t,compared to their integer-order analogues[57,60,68]and attracted exten-sive research(see,e.g.,[7,16,17,21,22,28,35,36,38,39,47,56,75,86]).The majority of the works in the literature focused on FPDEs modeling anomalously diffusive transport,including both subdiffusive transport and superdiffusive transport or a combination of both.One can find that an implicit finite difference approximation to space-time fractional advection equations(FAEs),obtained by directly truncating the Grunwald-Letnikov fractional derivative,is unstable as proved in[59].The reason is that the numerical approximation is uniquely determined by only the left-sided boundary condition while the solution to the continuous problem can only be uniquely determined with the boundary condition at both the endpoints of the interval.Unlike fractional diffusion problems,the two boundary conditions in the two-sided FAE problem work against each other in the sense that the two fronts generated by the two boundary conditions move towards each other as in the turning point problems in integer-order advection-diffusion equations that tend to generate numerical approximations with spurious oscillations and under and over shoot as well as other numerical difficulties[76].Inspired by these considerations,we follow the ideas in[50,99,100,101,103]to develop a bound-preserving numerical method for space-fractional and space-time fractional advection equations,so the resulting numerical approximations can remove spurious numerical oscillation and under and over shoot.Because fractional differential operators involve singular integral opera-tors,numerical methods for space-fractional PDEs usually generate dense or full stiffness matrices.In addition,due to the memory effect of the time-fractional differential operator.the numerical discretizations of time-fractional PDEs generate numerical schemes that involve the numerical solu-tions at all the previous time steps.Fast numerical methods were developed for the efficient numerical simulations of space-fractional PDEs in multiple space dimensions,which are expressed in terms of fractional differential op-erators in the coordinate directions,by utilizing the Toeplitz-like structure of their stiffness matrices and the tensor product structure of the stiffness matrices,which reduce the memory requirement from O(N2)to O(N)and computational complexity from O(N3)to O(N log N)per Krylov subspace iteration[86,87,88].In general,the fractional PDEs will be in the form of directional fractional derivatives.Consequently,the stiffness matrix of the corresponding finite element discretization has a much more complicated structure than its counterpart for the space-fractional PDEs in the coordinate form[75].In other words,the spatially directional fractional PDEs model more general anomalously diffusive transport processes than the coordinate form space fractional PDEs do physically and are more difficult to handle.The main contents of this thesis is arranged as follows:In chapter 1,we introduce briefly the history and some basic definitions of the fractional derivative and integral operators as well as some properties.Then we show some special matrices which are related to the fast methods in the thesis.In chapter 2 and chapter 3,we provide an explicit finite difference method(EFDM)for the space-fractional PDEs and space-time fractional PDEs in one dimension and in two dimensions,respectively.We prove its boundary-preserving principle and error estimate.Fast implementations are also derived which significantly reduce the computational complexity from O(MN3+M2N)to O(MNlog(MN))and the memory requirement from O(MN2)to O(NlogM).we carry out numerical experiments to substan-tiate the theoretical analysis and investigate the performance of the fast implementation.In chapter 4,we analyze a fast finite element method for space-time fractional PDEs with spatially directional fractional derivatives in space and prove its error estimate and present its numerical implementation which sig-nificantly reduces the computational complexity of the numerical method from O(MN3+M2N)to O(MN log(MN))per iteration in the fast method.Furthermore,the fast implementation reduces the memory requirement from O(MN2)to O(N log M).Numerical experiments are carried out to sub-stantiate the theoretical analysis and investigate the performance of the fast implementation.