Preconditioned Methods For Several Classes Of Space Fractional Diffusion Equations

Fractional differential equations,as generalizations of integer-order differential equations,have attracted considerable attention of researchers in recent decades.On the one hand,this can be attributed to the self-development of theories of fractional calculus.On the other hand,this is because fractional differential equations have been successfully applied to many research fields,especially in characterizing the anomalous diffusion process.As the closed solution is hard to obtain,it is significant to numeri-cally solve the fractional equations.However,due to the nonlocal property of fractional operators,the coefficient matrices of the linear equations obtained by discretizing the fractional equations are typically dense and usually ill-conditioned.This brings big dif-ficulties and challenges in designing efficient numerical solvers.In this dissertation,we focus on the discretized linear equations of several classes of typical Spatial Fractional Diffusion Equations(SFDEs),and study designing efficient preconditioners so that the corresponding preconditioned Krylov subspace iteration methods can converge rapidly.The main works are as follows:(1)For steady-state SFDEs,we study two kinds of diagonal-times-Toeplitz pre-conditioners P_Tand P_H,as well as their corresponding inexact variants which can be implemented efficiently.For the SFDEs whose diffusion coefficients satisfying certain proportional relationships,we analyze the clustering properties of the spectra of the co-efficient matrices preconditioned by P_Tand P_H.For general SFDEs,we utilize the theory of GLT sequences to analyze the clustering properties of the singular values of the preconditioned coefficient matrices.Meanwhile,we prove that the coefficient ma-trices preconditioned by the inexact P_Tand P_Halso have the clustering properties.The results suggest that P_Tand its inexact variant are effective when one of the diffusion coefficients is significantly larger than the other one,whereas P_Hand its inexact vari-ant are effective when the fractional order is close to 2 or the two diffusion coefficients are relatively close to each other.Numerical results show that the inexact P_Tand P_Hcan make the preconditioned GMRES method converge rapidly and,compared with the other tested preconditioners,the two new preconditioners are more efficient.(2)For unsteady-state SFDEs,we first approximate the Toeplitz matrices involved in the discretized SFDEs by circulant matrices,and then design a kind of Diagonal-Times-Circulant Splitting(DTCS)preconditioner based on the idea of matrix splitting and the skill of alternating direction.We analyze the clustering property of the spectra of the preconditioned coefficient matrices.The results indicate that the DTCS precon-ditioner will be effective when one of the diffusion coefficients is sufficiently larger than the other one and the maximum and minimum values of the larger coefficients are relatively close to each other.Numerical results show that the DTCS-preconditioned GMRES method can converge rapidly,and the overall performances of the DTCS pre-conditioner are more effective and robust than the other tested preconditioners.(3)For conservative SFDEs,by further generalizing the method for designing the DTCS preconditioner,we propose a kind of Approximate Splitting(AS)preconditioner.We analyze the clustering property of the spectra of the preconditioned coefficient matri-ces.The results indicate that the AS preconditioner will be effective when the maximum and minimum values of the diffusion coefficients are relatively close to each other.Nu-merical results show that the AS-preconditioned GMRES method can converge rapidly,and the overall performances of the AS preconditioner are superior to other tested pre-conditioners.(4)For balanced SFDEs,based on the idea of approximate inverse and the tech-nique of interpolation approximation,we design a Symmetric Circulant Approximate Inverse(SCAI)preconditioner which can be used in the preconditioned CG method.Theoretical results indicate that the eigenvalues of the preconditioned coefficient ma-trices are clustered around 1 when the diffusion coefficient is uniformly continuous.Meanwhile,we also take the technique of low rank correction to further improve the pre-conditioning effect of the SCAI preconditioner and design a Corrected SCAI(CSCAI)preconditioner.Numerical results show that the SCAI preconditioner can make the pre-conditioned CG method converge rapidly and the CSCAI preconditioner can be more efficient than the SCAI preconditioner for the one-dimensional problems.The overall performances of the SCAI and CSCAI preconditioners are superior to the other tested preconditioners.