A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains

Abstract Fractional partial differential equations (PDEs) provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long-range spatial interactions, which cannot be modeled accurately by classical second-order diffusion equations. However, numerical methods for space-fractional PDEs usually generate dense or full stiffness matrices, for which a direct solver requires O ( N 3 ) computations per time step and O ( N 2 ) memory, where N is the number of unknowns. The significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three-dimensional space-fractional diffusion equations computationally intractable. Fast numerical methods were previously developed for space-fractional PDEs on multidimensional rectangular domains, without resorting to lossy compression, but rather, via the exploration of the tensor-product form of the Toeplitz-like decompositions of the stiffness matrices. In this paper we develop a fast finite difference method for distributed-order space-fractional PDEs on a general convex domain in multiple space dimensions. The fast method has an optimal order storage requirement and almost linear computational complexity, without any lossy compression. Numerical experiments show the utility of the method.

[1]  Fawang Liu,et al.  Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains , 2015 .

[2]  Hong Wang,et al.  Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations , 2014, J. Comput. Phys..

[3]  Hong Wang,et al.  A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations , 2013, J. Comput. Phys..

[4]  Hong Wang,et al.  A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation , 2013, J. Comput. Phys..

[5]  V. Ervin,et al.  Variational solution of fractional advection dispersion equations on bounded domains in ℝd , 2007 .

[6]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

[7]  Hong Wang,et al.  A Fast Finite Element Method for Space-Fractional Dispersion Equations on Bounded Domains in ℝ2 , 2015, SIAM J. Sci. Comput..

[8]  K. Burrage,et al.  Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain , 2012 .

[9]  Hong Wang,et al.  A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh , 2015, J. Comput. Phys..

[10]  Xiao-Qing Jin,et al.  Preconditioned iterative methods for space-time fractional advection-diffusion equations , 2015, J. Comput. Phys..

[11]  Siu-Long Lei,et al.  A circulant preconditioner for fractional diffusion equations , 2013, J. Comput. Phys..

[12]  E. Montroll,et al.  Anomalous transit-time dispersion in amorphous solids , 1975 .

[13]  Hong Wang,et al.  Fast preconditioned iterative methods for finite volume discretization of steady-state space-fractional diffusion equations , 2016, Numerical Algorithms.

[14]  H. R. Hicks,et al.  Numerical methods for the solution of partial difierential equations of fractional order , 2003 .

[15]  Hong Wang,et al.  An O(N log2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations , 2011, J. Comput. Phys..

[16]  Hong Wang,et al.  A direct O(N log2 N) finite difference method for fractional diffusion equations , 2010, J. Comput. Phys..

[17]  David A. Benson,et al.  Space‐fractional advection‐dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data , 2007 .

[18]  Hong Wang,et al.  A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations , 2012, SIAM J. Sci. Comput..

[19]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .

[20]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[21]  Aijie Cheng,et al.  A preconditioned fast Hermite finite element method for space-fractional diffusion equations , 2017 .

[22]  Fawang Liu,et al.  A numerical investigation of the time distributed-order diffusion model , 2014 .

[23]  Vickie E. Lynch,et al.  Fractional diffusion in plasma turbulence , 2004 .

[24]  Hong Wang,et al.  Fast Iterative Solvers for Linear Systems Arising from Time-Dependent Space-Fractional Diffusion Equations , 2016, SIAM J. Sci. Comput..

[25]  Robert M. Gray,et al.  Toeplitz and Circulant Matrices: A Review , 2005, Found. Trends Commun. Inf. Theory.

[26]  D. Benson,et al.  The fractional‐order governing equation of Lévy Motion , 2000 .

[27]  M. Meerschaert,et al.  Finite difference methods for two-dimensional fractional dispersion equation , 2006 .

[28]  Michael K. Ng,et al.  Preconditioning Techniques for Diagonal-times-Toeplitz Matrices in Fractional Diffusion Equations , 2014, SIAM J. Sci. Comput..

[29]  I. M. Sokolov,et al.  Distributed-Order Fractional Kinetics , 2004 .

[30]  Rudolf Hilfer,et al.  On fractional diffusion and continuous time random walks , 2003 .