High-order Numerical Methods For Fractional Diffusion Equations

Fractional differential equations have been widely used to describe the complex dynamic problems with memory and heredity.Since the fractional derivative is nonlocal,only a few simple fractional differential equations can be solved by analytical methods.This makes the investigation of numerical methods for the fractional differential equations become an urgent and important subject.The dissertation is concerned with constructing high order numerical methods for the Riesz space fractional and distributed-order diffusion equations,and giving the stability and convergence of this methods.The main work of the dissertation contains four parts as follows:A numerical method with high order both in time and in space is constructed for the Riesz space fractional diffusion equation,the idea of construction is to discretize the temporal term and the spatial term by the s-stage implicit Runge-Kutta method and the spectral Galerkin method,respectively.For an algebraically stable Runge-Kutta method of order p(p ? s + 1),the stability of this method is proven and the convergence order of s + 1 in time is obtained.By using the regularity estimate of solution,the optimal error estimate in space,with convergence order only depending on the regularity of initial value and right hand function,is also derived.Moreover,combining with the high accuracy Gauss-Legendre quadrature formula,this kind of method is extended to the Riesz space distributed-order diffusion equation,and similar stability and convergence results are obtained.By introducing the k-step backward difference formula in the time direction,and applying the spectral Galerkin method in the space direction,a numerical method with low computation cost and high order both in time and space is proposed for the Riesz space fractional diffusion equation.This method avoids the problem that the implicit Runge-Kutta method has expensive computation cost.By using the G-theory and the multiplier technique,the stability of this method is proven and the convergence order of k(k ? 5)in time is obtained.The optimal error estimate in space is also derived.Moreover,this kind of method is extended to the two dimensional Riesz space fractional diffusion equation,and the corresponding stability and convergence results are obtained.By using the general linear method,which is a general formulation of a series classical methods,and the spectral Galerkin method,a more general numerical method with high order both in time and space is further constructed for the Riesz space fractional diffusion equation.For an irreducible and algebraically stable general linear method with generalized stage order p,the stability of this method is proven and the convergence order of p in time is obtained.The optimal error estimate in space is also derived.For the more complex two-dimensional nonlinear Riesz space fractional diffusion equation,by using the s-stage implicit Runge-Kutta method and the spectral Galerkin method,a numerical method with high order both in time and in space is constructed.For an algebraically stable and coercive s-stage Runge-Kutta method,if the nonlinear function fulfils the Lipschitz condition,the stability of this method is proven.When the s-stage Runge-Kutta method has order p(p ? s + 1),the convergence order of s + 1 in time of this method is proven,and the optimal spatial error estimate with convergence order independent of the regularity of solution is derived.Moreover,this kind of method is used to solve the two-dimensional nonlinear Riesz space distributed-order diffusion equation,and similar stability and convergence results are obtained,where the distributed-order is discretized by the Gauss-Legendre quadrature formula.