Finite Difference Methods For Space Fractional Differential Equations And Fast Algorithms

Fractional differential equations has attracted considerable attention during the past two decades,due mainly to its broad applications in widespread fields of science and engineering,such as turbulent flow,groundwater contaminant transport,material,biological systems,signal and image processing.In the thesis,we consider the finite difference methods and the corresponding fast algorithms for the numerical solutions of space fractional differential equations.The main contents of this thesis are in Chapters 2-7.In Chapter 2,we introduce 3-point WSGD and 4-point WSGD operators which are used to approximate the Riemann-Liouville fractional derivatives of order α∈(0,1)and α∈(1,2)respectively,including error estimations.At the same time,by using extrapolation technique for 3-point WSGD operator,the extrapolated WSGD(EWSGD)operator with fourth order accuracy is also given.In Chapter 3,we consider the finite difference methods for the solutions of the one-and two-dimensional space fractional diffusion equations with constant coefficients.Firstly,the QCD operator is used to approximate the RimeannLiouville fractional derivatives,which results in a system of ordinary differential equations(ODEs).Then the implicit Runge-Kutta method is applied to discretize the resulted ODEs.Finally,an IRK-QCD scheme with temporally and spatially fourth order accuracy is proposed,and the stability and the convergence of the IRK-QCD scheme are theoretically established.In Chapter 47 we consider the finite difference methods for the solutions of the one-and two-dimensional space fractional diffusion equations with variable coefficients.The 4-point WSGD operator introduced in Chapter 2 is applied to discretize the Riemann-Liouville fractional derivatives,and the Crank-Nicolson technique is used to discretize the temporal derivative,which results in CN4WSGD schemes.By choosing different parameters,a CN-4WSGD scheme can achieve spatial third or fourth order accuracy.Finally,we theoretically analyze the stability and the convergence of CN-4WSGD schemes.In Chapter 5,we consider finite difference methods for the solutions of the one-dimensional space fractional advection-diffusion equations with variable coefficients.We use 3-point WSGD operators with different parameters to approximate the fractional advection term and fractional diffusion term,respectively,then a CN-WSGD scheme is obtained.The stability and convergence of the CN-WSGD scheme are discussed in four cases according to the dependence of coefficients on time and space variables.Meanwhile,we discuss the choice of the parameters and propose a class of unconditionally stable CN-WSGD schemes.Numerical results are implemented to verify the theoretical results.In Chapters 6-7,we study the discretization linear systems discussed in Chapters 2-4,and propose fast algorithms for the linear systems.For the linear systems of IRK-QCD scheme,we construct a block-lower-triangular(BLT)preconditioner and prove that the condition number of the preconditioned matrix is uniformly bounded and the eigenvalues of the preconditioned matrix lie in disc ｛λ:λ∈C,|λ-1|≤1/3}.For the linear systems of CN-4WSGD schemes,we propose diagonal times Toeplitz splitting(DTTS)iteration by splitting the coefficient matrix,and prove the uniform convergence without imposing any extra conditions.Whereafter,we discuss fast algorithms for Riesz space fractional diffusion equations with variable coefficients.We first transform the asymmetric linear system into symmetric linear system,then the banded preconditioner with diagonal compensation is proposed.And we prove that the eigenvalues of the preconditioned matrix are clustered around 1.Finally,the accuracy of the IRK-QCD and CN-4WSGD schemes,and the effectiveness of the fast algorithms are verified by numerical results.